# Part 5: Sequences and Series

$$\newcommand{\<}{\langle} \newcommand{\>}{\rangle}$$

## 5.1 Sequences

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 9.1

### 5.1.1 Definition

• A sequence is an infinitely long list of real numbers. For example, the sequence of positive even integers is $$\<2,4,6,8,\dots\>$$.
• Example Use your intuition to guess the next three terms of the sequences $$\<1,3,5,7,9,\dots\>$$, $$\<3,-6,9,-12,15,\dots\>$$, and $$\<0,1,4,9,16,\dots\>$$.
• An explicit formula $$a_n$$ is a rule defining each term of the sequence, where $$n=0$$ yields the first term, $$n=1$$ gives the next term, and so on. The sequence generated by the formula $$a_n$$ is written as $$\<a_n\>_{n=0}^\infty=\<a_0,a_1,a_2,\dots\>$$.
• Occasionally the first term of the sequence may be given by an integer different from $$0$$, in which case the sequence is written like $$\<a_n\>_{n=1}^\infty$$.
• Example Write the first five terms of the sequences $$\<a_n\>_{n=0}^\infty$$, $$\<b_n\>_{n=0}^\infty$$, and $$\<c_n\>_{n=0}^\infty$$ defined by $$a_n=4n$$, $$b_n=\frac{(-1)^n}{n^2+2}$$, and $$c_n=\cos(\frac{\pi}{2}n)$$.
• Example Give the term $$a_7$$ for the sequence defined by the formula $$a_n=\frac{n}{2n+1}$$.

### 5.1.2 Recursive Formulas

• A recursive formula for a sequence defines one or more initial terms of the sequence, and then defines future terms of the sequence by using previous terms.
• Example Write the first ten terms of the Fibonacci sequence defined by the recursive formula $$f_0=1,f_1=1,f_{n+2}=f_n+f_{n+1}$$.
• Example Write the first six terms of the factorial sequence defined by the recursive formula $$!_0=1,!_{n+1}=(n+1)!_n$$.
• The factorial sequence is commonly written in the form $$n!$$ rather than $$!_n$$. It has the explicit formula $$n!=1\times2\times3\times\dots\times n$$.
• Example Prove that $$a_n=\frac{3}{2^n}$$ is an explicit formula for the sequence $$\<a_n\>_{n=0}^\infty$$ defined recursively by $$a_0=3,a_{n+1}=\frac{a_n}{2}$$.

### 5.1.3 Limits, Convergence, and Divergence

• The sequence $$\<a_n\>_{n=i}^\infty$$ converges to a limit $$L$$ if for each $$\epsilon>0$$, there exists an integer $$N$$ such that $$|a_n-L|<\epsilon$$ for all $$n\geq N$$. This is written as $$\lim_{n\to\infty}a_n=L$$ or $$a_n\to L$$.
• Example Guess the limit of the harmonic sequence
$$\<a_n\>_{n=1}^\infty$$ defined by $$a_n=\frac{1}{n}$$ by writing out the first few terms.
• Example Guess the limit of the sequence
defined by $$g_n=\frac{2^n}{2^{n+1}}$$ by writing out the first few terms.
• A sequence diverges when it doesn’t converge to any limit.
• Example Write a few terms of the sequence defined by the formula $$b_n=(-1)^n\frac{n+1}{n+2}$$. Does it appear to be converging or diverging?

### Exercises for 5.1

1. Use your intuition to guess the next three terms of the sequences $$\<1,5,9,13,17,\dots\>$$, $$\<1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\frac{1}{25},\dots\>$$, and $$\<\frac{1}{3},-1,3,-9,27,\dots\>$$.
2. Create an explicit formula for each of the three previous sequences.
3. Write the first five terms of the sequences $$\<a_n\>_{n=0}^\infty$$, $$\<b_n\>_{n=0}^\infty$$, and $$\<c_n\>_{n=0}^\infty$$ defined by $$a_n=3n+2$$, $$b_n=2(-\frac{1}{3})^n$$, and $$c_n=\frac{n}{1+n^2}$$.
4. Write the first six terms of the sequence $$\<q_n\>_{n=0}^\infty$$ defined by $$q_0=0$$ and $$q_{n+1}=q_n+2n+1$$.
5. Prove that $$q_n=n^2$$ is an explicit formula for the sequence defined recursively in the previous problem.
6. Write the first six terms of the sequence $$\<b_n\>_{n=1}^\infty$$ defined by $$b_1=4$$ and $$b_{n+1}=\frac{b_n}{2}$$.
7. Prove that $$b_n=\frac{8}{2^n}$$ is an explicit formula for the sequence defined recursively in the previous problem.
8. Guess the limit of the alternating harmonic sequence
$$\<b_n\>_{n=1}^\infty$$ defined by $$b_n=\frac{(-1)^n}{n}$$ by writing out the first few terms.
9. Guess the limit of the geometric sequence
$$\<g_n\>_{n=0}^\infty$$ defined by $$g_n=2^{-n}$$ by writing out the first few terms.
10. Guess the limit of the sequence
$$\<a_n\>_{n=3}^\infty$$ defined by $$a_n=\frac{3n+2}{2n+1}$$ by writing out the first few terms.
11. Write a few terms of the sequence defined by the formula $$c_n=\frac{n!}{n^2+1}$$. Does it appear to be converging or diverging?
12. Write a few terms of the sequence defined by the formula $$s_n=\sin(\frac{\pi n}{3})$$. Does it appear to be converging or diverging?
13. (OPTIONAL) Sketch a picture which explains why $$\lim_{n\to\infty} \sin(\pi n)=0$$ as the limit of a sequence, but $$\lim_{x\to\infty}\sin(\pi x)$$ does not exist as a limit of a function.
14. (QUIZ) What are the first five terms of the sequence $$\<r_n\>_{n=1}^\infty$$ defined explicitly by $$r_n=\frac{n+2}{3+n^2}$$?
• $$\<\frac{3}{4},\frac{4}{7},\frac{5}{12},\frac{6}{19}, \frac{1}{4},\dots\>$$
• $$\<\frac{2}{7},\frac{1}{2},\frac{4}{9},0,\frac{5}{17},\dots\>$$
• $$\<0,\frac{3}{5},\frac{5}{18},\frac{8}{27},\frac{9}{61},\dots\>$$
15. (QUIZ) What are the first five terms of the sequence $$\<w_n\>_{n=0}^\infty$$ defined recursively by $$w_0=1$$, $$w_1=2$$, $$w_{n+2}=2w_n+w_{n+1}$$?
• $$\<1,2,5,10,17,\dots\>$$
• $$\<1,2,3,5,9,\dots\>$$
• $$\<1,2,4,8,16,\dots\>$$
16. (QUIZ) Which of these statements seems most appropriate for describing the sequence whose initial terms are $$\<1,\frac{3}{4},\frac{5}{8},\frac{9}{16},\frac{17}{32},\dots\>$$?
• The sequence appears to converge to $$\frac{1}{2}$$.
• The sequence appears to diverge to $$\frac{1}{2}$$.
• The sequence appears to neither converge nor diverge.

Solutions 1-7

Solutions 8-16

## 5.2 Computing Limits of Sequences

• University Calculus: Early Transcendentals (3rd Ed)
• 9.1

### 5.2.1 Limits of Sequences and Functions

• If $$f(x)$$ is a function and $$a_n$$ is a sequence such that $$f(n)=a_n$$ for sufficiently large integers $$n$$, then $$\lim_{x\to\infty}f(x)=L$$ implies $$\lim_{n\to\infty}a_n=L$$.
• Therefore all the rules for evaluating $$\lim_{x\to\infty}f(x)$$ extend to evaluating $$\lim_{n\to\infty}a_n$$.
• Example Use factoring to compute $$\lim_{n\to\infty}\frac{4+n}{n^3+1}$$.
• Example Use L’Hopital’s Rule to prove that any sequence defined by the formula $$a_n=\frac{n^2+3}{4-5n^2}$$ converges to $$-\frac{1}{5}$$.
• Example Use the squeeze theorem to compute $$\lim_{n\to\infty}\frac{\sin n}{n}$$.

### 5.2.2 Common Limits

• The following limits are often useful:
• $$\lim_{n\to\infty} x = x$$
• $$\lim_{n\to\infty} \frac{1}{n} = 0$$
• $$\lim_{n\to\infty} \frac{\ln n}{n} = 0$$
• $$\lim_{n\to\infty} \sqrt[n]{p(n)} = 1$$ where $$p(n)$$ is a polynomial
• $$\lim_{n\to\infty} x^n = 0$$, $$|x|<1$$
• $$\lim_{n\to\infty} (1+\frac{x}{n})^n=e^x$$
• $$\lim_{n\to\infty} \frac{x^n}{n!}=0$$
• Example Find $$\lim_{n\to\infty}\frac{\ln(n^3)}{n}$$.
• Example Find $$\lim_{n\to\infty}\frac{3^n+1}{n!}$$.
• Example Find $$\lim_{n\to\infty}(4n)^{1/n}$$.

### 5.2.3 Monotonic and Bounded Sequences

• A sequence $$\<a_n\>_{n=i}^\infty$$ is bounded if there exist real numbers $$A,B$$ such that $$A\leq a_n\leq B$$ for all integers $$n\geq i$$.
• Example Is the sequence $$\<a_n\>_{n=1}^\infty$$ where $$a_n=\frac{n+1}{n}$$ bounded?
• Example Is the sequence $$\<b_n\>_{n=0}^\infty$$ given by $$b_n=\frac{n}{(-3)^n}$$ bounded?
• A sequence is monotonic if it either never increases or never decreases.
• Example Is the sequence $$\<a_n\>_{n=1}^\infty$$ where $$a_n=\frac{n+1}{n}$$ monotonic?
• Example Is the sequence $$\<b_n\>_{n=0}^\infty$$ given by $$b_n=\frac{n}{(-3)^n}$$ monotonic?
• The Monotonic Sequence Theorem states that all bounded monotonic sequences converge.

### Exercises for 5.2

1. Use factoring to compute $$\displaystyle\lim_{n\to\infty}\frac{n-4n^2}{2n^2+7}$$.
2. Use L’Hopital’s Rule to prove that $$\displaystyle\frac{\ln n}{n}\to 0$$.
3. Use the squeeze theorem to compute $$\displaystyle\lim_{n\to\infty}\frac{\cos n}{n\ln n}$$.
4. Find $$\displaystyle\lim_{n\to\infty}\frac{\sin n + 3n^2}{n^2+1}$$.
5. Find $$\displaystyle\lim_{n\to\infty}\frac{\ln(n^n)}{n^2}$$.
6. Find $$\displaystyle\lim_{n\to\infty}(5n^3)^{2/n}$$.
7. Find $$\displaystyle\lim_{n\to\infty}(\frac{1}{\pi})^{3n}$$.
8. Find $$\displaystyle\lim_{n\to\infty}(\frac{1}{2}+\frac{1}{n})^n$$.
9. Find $$\displaystyle\lim_{n\to\infty} \frac{\frac{(n+2)!}{2^n}}{\frac{3n^2n!}{2^{n+1}}}$$.
10. Based on its first few terms, does the sequence $$\<a_n\>_{n=2}^\infty$$ where $$a_n=\frac{2+n^2}{n^2-1}$$ appear bounded? Monotonic? Does it appear to converge?
11. Based on its first few terms, does the sequence $$\<b_n\>_{n=0}^\infty$$ where $$b_n=(-3)^n$$ appear bounded? Monotonic? Does it appear to converge?
12. Based on its first few terms, does the sequence $$\<y_n\>_{n=1}^\infty$$ where $$y_n=(-\frac{1}{2})^n$$ appear bounded? Monotonic? Does it appear to converge?
13. (OPTIONAL) Prove that $$\displaystyle\lim_{n\to\infty} (1+\frac{x}{n})^n=e^x$$ by considering the function version $$\displaystyle L=\lim_{t\to\infty} (1+\frac{x}{t})^t$$ and taking the natural log of both sides of the equality. Use L’Hopital to solve this limit, showing that $$\ln L=x$$ and therefore $$L=e^x$$.
14. (QUIZ) Find $$\lim_{n\to\infty}\frac{n!\cos n}{(n+1)!}$$.
• $$1$$
• $$0$$
• $$\pi/2$$
15. (QUIZ) Find $$\lim_{n\to\infty}\frac{(3+n)^n}{n^n}$$.
• $$1$$
• $$0$$
• $$e^3$$
16. (QUIZ) Which of these statements seems most appropriate for describing the sequence whose initial terms are $$\<\frac{1}{4},-\frac{1}{6},\frac{1}{8},-\frac{1}{10}, \frac{1}{12},\dots\>$$?
• The sequence is bounded and monotonic, so it converges by the Monotonic Sequence Theorem.
• The sequence is not monotonic and not bounded, so it diverges by the Monotonic Sequence Theorem.
• The sequence is bounded, but not monotonic, so the Monotonic Sequence Theorem doesn’t apply. However, it does appear to converge to $$0$$ anyway.

Solutions 1-13

Solutions 14-16

## 5.3 Series

• University Calculus: Early Transcendentals (3rd Ed)
• 9.2

### 5.3.1 Series as Partial Sum Sequences

• For a given sequence $$\<a_n\>_{n=0}^\infty$$, its partial sum sequence $$\<s_n\>_{n=0}^\infty$$ is defined explicitly by $$s_n=\sum_{i=0}^n a_i=a_0+a_1+\dots+a_n$$, and defined recursively by $$s_0=a_0$$ and $$s_{n+1}=s_n+a_{n+1}$$.
• Example Write out the first few terms of the partial sum sequence for $$\<1,2,3,4,5,\dots\>$$.
• Example Write out the first few terms of the partial sum sequence for $$\<b_i\>_{i=1}^\infty$$ where $$b_i=\frac{6}{i}$$.
• The series $$\sum_{n=0}^\infty a_n=a_0+a_1+a_2+\dots$$ represents the sum of the infinite sequence $$\<a_n\>_{n=0}^\infty$$. If its partial sum sequence converges to $$L$$, then we say that its series converges to $$L$$ and the value of the series is $$L$$ (written $$\sum_{n=0}^\infty a_n=a_0+a_1+a_2+\dots=L$$). Otherwise, we say the series diverges.

### 5.3.2 Telescoping/Geometric Sequences and Series

• A telescoping series is a series whose partial sum sequence allows for canceling.
• Example Show that $$\sum_{n=1}^\infty(\frac{1}{n}-\frac{1}{n+1})$$ converges to $$1$$ by evaluating the limit of its partial sum sequence.
• Example Does $$\sum_{n=0}^\infty\frac{2}{n^2+3n+2}$$ converge or diverge?
• The geometric series defined for real numbers $$a,r$$ is $$\sum_{n=0}^\infty ar^n=a+ar+ar^2+ar^3+\dots$$.
• The geometric series $$\sum_{n=0}^\infty ar^n$$ converges to $$\frac{a}{1-r}$$ when $$|r|<1$$, and diverges when $$|r|\geq 1$$.
• Example Compute $$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots$$.
• Example Does $$\sum_{k=0}^\infty\frac{2}{3^{k+1}}$$ converge or diverge? If it converges, what is its value?
• Example Does $$\sum_{k=0}^\infty\frac{2}{(1/3)^{k+1}}$$ converge or diverge? If it converges, what is its value?

### 5.3.3 Divergent Series

• The Series Divergence Test: If a sequence fails to converge to $$0$$, then its series diverges.
• Example Does $$\sum_{k=0}^\infty\frac{k^2+3}{2k^2+k+5}$$ converge or diverge? If it converges, what is its value?
• This does NOT mean that if a sequence converges, then its series converges.
• The harmonic sequence $$\<\frac{1}{n}\>_{n=1}^\infty$$ converges to $$0$$, but its series $$\sum_{n=1}^\infty\frac{1}{n}$$ diverges.

### 5.3.4 Arithmetic Rules and Reindexing

• Because a series is a limit, it follows the same rules as limits do.
• Example Evaluate the convergent series $$\sum_{i=0}^\infty\frac{1+\frac{2^{i+2}}{i+1}-\frac{2^{i+2}}{i+2}}{2^i}$$.
• The starting index for a series may be adjusted by offsetting the index for its sequence in the opposite direction.
• Example Does $$\sum_{m=-1}^\infty\frac{1}{m+2}$$ converge or diverge? If it converges, what is its value?

### Exercises for 5.3

1. Write out the first four terms of the partial sum sequence for $$\<1,-\frac{1}{3},\frac{1}{9},-\frac{1}{27},\dots\>$$.
2. Write out the first four terms of the partial sum sequence for $$\<0.3,0.03,0.003,0.0003,\dots\>$$.
3. Does $$\sum_{m=2}^\infty(\frac{3}{2m}-\frac{3}{2m+2})$$ converge or diverge? If it converges, what is its value?
4. Does $$\sum_{j=2}^\infty\frac{6}{4j^2+4j}$$ converge or diverge? If it converges, what is its value?
5. Compute $$1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\dots$$.
6. Prove that $$0.\overline3=0.333\dots$$ equals $$\frac{1}{3}$$ by expressing the decimal expression as a geometric series.
7. Write $$0.\overline{27}=0.272727\dots$$ as a fraction of integers.
8. Does $$\sum_{n=0}^\infty\frac{6}{3^{n+2}}$$ converge or diverge? If it converges, what is its value?
9. Does $$\sum_{m=0}^\infty 3(-1)^m$$ converge or diverge? If it converges, what is its value?
10. Does $$\sum_{i=1}^\infty \frac{i+\sin i}{2i}$$ converge or diverge? If it converges, what is its value?
11. Suppose $$\sum_{n=0}^\infty a_n=3$$ and $$\sum_{n=0}^\infty b_n=4$$. Evaluate $$\sum_{n=0}^\infty(3a_n-2b_n)$$.
12. Does $$\sum_{k=2}^\infty 4(\frac{2}{3})^k$$ converge or diverge? If it converges, what is its value?
13. (OPTIONAL) Prove $$\sum_{n=1}^\infty\frac{1}{3^n}=\frac{1}{2}$$ using the proof of the Geometric Series formula (not the formula itself).
14. (QUIZ) Does $$\sum_{n=3}^\infty\left(\frac{6}{n}-\frac{6}{n+1}\right)$$ converge or diverge? If it converges, what is its value?
• It converges to $$\frac{1}{2}$$.
• It converges to $$2$$.
• It diverges.
15. (QUIZ) Does $$\sum_{i=0}^\infty\frac{(-3)^i}{2}$$ converge or diverge? If it converges, what is its value?
• It converges to $$-3$$.
• It converges to $$6$$.
• It diverges.
16. (QUIZ) Does $$\sum_{n=1}^\infty\frac{1}{4^n}$$ converge or diverge? If it converges, what is its value?
• It converges to $$\frac{1}{3}$$.
• It converges to $$\frac{3}{4}$$.
• It diverges.

Solutions 1-7

Solutions 8-16

## 5.4 The Integral Test

• University Calculus: Early Transcendentals (3rd Ed)
• 9.3, 8.7

### 5.4.1 Improper Integrals

• If $$f(x)\geq 0$$, the improper integral $$\int_a^\infty f(x)\,dx=\lim_{b\to\infty}\int_a^b f(x)\,dx$$ represents the area under the curve $$y=f(x)$$ from $$x=a$$ out to $$\infty$$. If the limit exists, then the improper integral converges; otherwise it diverges.
• Example Does $$\int_1^\infty\frac{1}{x^2}\,dx$$ converge or diverge? If it converges, what is its value?
• Example Does $$\int_4^\infty\frac{1}{2\sqrt y}\,dy$$ converge or diverge? If it converges, what is its value?
• When an integrand is undefined at a bound of integration, then the integral is also called improper and is evaluated with a limit.
• Example Find the value of $$\int_0^8 z^{-1/3}\,dz$$.

### 5.4.2 The Integral Test

• If $$a_n=f(n)$$ where $$f(x)$$ is a continuous, positive, decreasing function for sufficiently large values of $$x$$, then the series $$\sum_{n=N}^\infty a_n$$ and improper integral $$\int_a^\infty f(x)\,dx$$ either both converge, or both diverge.
• Example Does $$\sum_{n=4}^\infty\frac{4n+4}{n^2+2n+1}$$ converge or diverge?
• Example Does $$\sum_{k=1}^\infty\frac{k}{e^{k^2}}$$ converge or diverge?
• Even when they both converge, the values of the series $$\sum_{n=N}^\infty a_n$$ and improper integral $$\int_N^\infty f(x)\,dx$$ usually differ.
• Example Show that $$\sum_{n=1}^\infty\frac{1}{n^3}\not=\int_1^\infty\frac{1}{x^3}\,dx$$.

### 5.4.3 The $$p$$-Series Test

• The $$p$$-Series Test states that the series $$\sum_{n=1}^\infty\frac{1}{n^p}$$ converges when $$p>1$$, and diverges when $$p\leq 1$$.
• Example Does $$\sum_{m=2}^\infty\frac{3}{\sqrt[10]{m^4}}$$ converge or diverge?
• Example Does $$\sum_{j=0}^\infty\frac{1}{j^2+2j+1}$$ converge or diverge?

### Exercises for 5.4

1. Does $$\int_2^\infty\frac{32}{x^3}\,dx$$ converge or diverge? If it converges, what is its value?
2. Does $$\int_0^\infty\frac{2y}{y^2+3}\,dy$$ converge or diverge? If it converges, what is its value?
3. Does $$\int_e^\infty\frac{1}{\ln(x^x)}\,dx$$ converge or diverge? If it converges, what is its value?
4. Show that $$\int_1^\infty\frac{1}{x^2}\,dx+1=\int_0^1\frac{1}{\sqrt y}\,dy$$. Then draw a sketch involving areas illustrating why they are equal.
5. Does $$\sum_{n=0}^\infty\frac{2n}{n^2+3}$$ converge or diverge?
6. Does $$\sum_{n=3}^\infty\frac{4}{n(\ln n)^3}$$ converge or diverge?
7. Does $$\sum_{n=-2}^\infty\frac{1}{e^n}$$ converge or diverge?
8. Show that $$\int_1^\infty\frac{1}{x^2}\,dx \not= \sum_{n=1}^\infty\frac{1}{n^2}$$, even though they both converge.
9. Does $$\sum_{k=100}^\infty\frac{5}{\sqrt[7]{k^6}}$$ converge or diverge?
10. Does $$\sum_{n=5}^\infty\frac{1}{n^2-8n+16}$$ converge or diverge?
11. (OPTIONAL) Does $$\sum_{n=-1}^\infty\frac{e^n}{1+e^{2n}}$$ converge or diverge? (Hint: $$\int\frac{1}{1+u^2}\,du=\tan^\leftarrow u+C$$ and $$\lim_{u\to\infty}\tan^\leftarrow u=\frac{\pi}{2}$$.)
12. (QUIZ) Does $$\sum_{m=0}^\infty\frac{2m}{(m^2+1)^2}$$ converge or diverge?
• It converges.
• It diverges.
• It both converges and diverges.
13. (QUIZ) Does $$\sum_{n=2}^\infty\frac{1}{\sqrt{n-1}}$$ converge or diverge?
• It converges.
• It diverges.
• It neither converges nor diverges.

Solutions 1-6

Solutions 7-13

## 5.5 Comparison Tests

• University Calculus: Early Transcendentals (3rd Ed)
• 9.4

### 5.5.1 Direct Comparison Test

• Suppose $$\sum_{n=N}^\infty a_n$$ is a series with non-negative terms.
• If there exists a convergent series $$\sum_{n=M}^\infty b_n$$ with non-negative terms where $$a_n\leq b_n$$ for sufficiently large $$n$$, then $$\sum_{n=N}^\infty a_n$$ converges as well.
• If there exists a divergent series $$\sum_{n=M}^\infty b_n$$ with non-negative terms where $$a_n\geq b_n$$ for sufficiently large $$n$$, then $$\sum_{n=N}^\infty a_n$$ diverges as well.
• Example Show that $$\sum_{n=0}^\infty\frac{n}{n^3+3n+2}$$ converges by comparing with the series $$\sum_{n=1}^\infty\frac{1}{n^2}$$.
• Following is a list of sequence formulas ordered from larger to smaller (for sufficiently large $$n$$).
• $$n^n$$
• $$n!$$
• $$b^n$$ where $$b>1$$ (such as $$2^n,e^n,10^n$$…)
• $$n^p$$ where $$p>0$$ (such as $$\sqrt{n},n,n^4$$…)
• $$\log_b n$$ where $$b>1$$ (such as $$\log_{10}(n),\ln(n),\log_2(n)$$…)
• any positive constant
• Example Does $$\sum_{n=1}^\infty\frac{2}{n^{1/3}+5}$$ converge or diverge?
• Example Does $$\sum_{k=3}^\infty\frac{3^n}{n!}$$ converge or diverge?
• Example Does $$\sum_{m=2}^\infty(m\ln m)^{-1/2}$$ converge or diverge?

### 5.5.2 Limit Comparison Test

• Suppose $$\sum_{n=N}^\infty a_n$$ is a series with non-negative terms.
• If there exists a convergent series $$\sum_{n=M}^\infty b_n$$ with non-negative terms where $$\lim_{n\to\infty}\frac{a_n}{b_n}<\infty$$, then $$\sum_{n=N}^\infty a_n$$ converges as well.
• If there exists a divergent series $$\sum_{n=M}^\infty b_n$$ with non-negative terms where $$\lim_{n\to\infty}\frac{a_n}{b_n}>0$$ (including divergence to infinity), then $$\sum_{n=N}^\infty a_n$$ diverges as well.
• Example Does $$\sum_{n=1}^\infty\frac{2}{n^{1/3}+5}$$ converge or diverge?
• Example Does $$\sum_{i=0}^\infty\frac{3i}{5^i}$$ converge or diverge?
• Example Does $$\sum_{n=42}^\infty\frac{2^n+5^n}{3^n+4^n}$$ converge or diverge?

### Exercises for 5.5

1. Does $$\sum_{n=0}^\infty\sqrt{\frac{n}{n^4+7}}$$ converge or diverge? (Use Direct Comparison.)
2. Does $$\sum_{n=3}^\infty\frac{4}{n^{0.8}-1}$$ converge or diverge? (Use Direct Comparison.)
3. Does $$\sum_{j=2}^\infty\frac{e^j}{e^{2j}+1}$$ converge or diverge? (Use Direct Comparison.)
4. Does $$\sum_{k=10}^\infty\frac{\sin^2(k)}{k^3}$$ converge or diverge? (Use Direct Comparison.)
5. Does $$\sum_{m=4}^\infty\frac{1}{\ln m}$$ converge or diverge? (Use Direct Comparison.)
6. Does $$\sum_{n=4}^\infty\frac{5}{2n+3}$$ converge or diverge? (Use Direct Comparison.)
7. Does $$\sum_{n=0}^\infty\sqrt{\frac{n}{n^4+7}}$$ converge or diverge? (Use Limit Comparison.)
8. Does $$\sum_{n=3}^\infty\frac{4}{n^{0.8}-1}$$ converge or diverge? (Use Limit Comparison.)
9. Does $$\sum_{j=2}^\infty\frac{e^j}{e^{2j}+1}$$ converge or diverge? (Use Limit Comparison.)
10. Does $$\sum_{k=10}^\infty\frac{\sin^2(k)}{k^3}$$ converge or diverge? (Use Limit Comparison.)
11. Does $$\sum_{m=4}^\infty\frac{1}{\ln m}$$ converge or diverge? (Use Limit Comparison.)
12. Does $$\sum_{n=4}^\infty\frac{5}{2n+3}$$ converge or diverge? (Use Limit Comparison.)
13. (OPTIONAL) Does $$\sum_{m=1}^\infty\frac{1}{1+2+\dots+(m-1)+m}$$ converge or diverge? (Hint: show that $$\frac{1}{1+2+\dots+(m-1)+m} = \frac{2}{(1+m)+(2+m-1)+\dots+(m-1+2)+(m+1)}$$.)
14. (QUIZ) Does $$\sum_{m=0}^\infty\frac{2m}{(m^2+1)^2}$$ converge or diverge?
• It converges.
• It diverges.
• It both converges and diverges.
15. (QUIZ) Does $$\sum_{n=1}^\infty\sqrt{\frac{n+1}{n^2+3}}$$ converge or diverge?
• It converges.
• It diverges.
• It neither converges nor diverges.

Solutions 1-8

Solutions 9-15

## 5.6 Absolute and Conditional Convergence

• University Calculus: Early Transcendentals (3rd Ed)
• 9.5, 9.6

### 5.6.1 Absolute and Conditional Convergence

• A series $$\sum_{n=N}^\infty a_n$$ absolutely converges whenever its absolute value series $$\sum_{n=N}^\infty |a_n|$$ converges. All absolutely convergent series converge normally.
• Example Show that $$\sum_{n=3}^\infty\frac{\cos n}{n^2}$$ absolutely converges.
• Example Show that $$\sum_{m=1}^\infty\frac{3^m}{(-4)^{m+1}}$$ absolutely converges.
• A convergent series which is not absolutely convergent is called conditionally convergent.
• Conditionally convergent series are named as such because the value of a conditionally convergent series depends on the order of its terms.

### 5.6.2 Alternating Series Test

• The Alternating Series Test: let $$\sum_{n=N}^\infty(-1)^n a_n$$ be a series such that $$\<a_n\>_{n=N}^\infty$$ has positive nonincreasing terms. Then $$\sum_{n=N}^\infty(-1)^n a_n$$ converges when $$\lim_{n\to\infty} a_n = 0$$.
• Also holds for $$\sum_{n=N}^\infty(-1)^{n\pm k} a_n$$
• Example Show that the alternating harmonic series $$\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}$$ is conditionally convergent.
• Example Is the series $$\sum_{k=3}^\infty\frac{\sin k}{k^2}$$ absolutely convergent, conditionally convergent, or divergent?
• Let $$a_n\geq0$$. Then the sequence $$\<(-1)^na_n\>_{n=N}^\infty$$ converges if and only if $$\<a_n\>_{n=N}^\infty$$ converges to zero.
• Example Is the series $$\sum_{n=0}^\infty\frac{(-e)^n}{n+1}$$ absolutely convergent, conditionally convergent, or divergent?
• Example Is the series $$\sum_{m=2}^\infty(-1)^m\frac{m}{m^{3/2} +3}$$ absolutely convergent, conditionally convergent, or divergent?

### Exercises for 5.6

1. Is the series $$\sum_{m=2}^\infty\frac{3}{1-m^2}$$ absolutely convergent, conditionally convergent, or divergent?
2. Is the series $$\sum_{k=1}^\infty\frac{\cos^5 k}{k^4}$$ absolutely convergent, conditionally convergent, or divergent?
3. Is the series $$\sum_{n=0}^\infty(-1)^{n+1}\frac{4}{n^2+3}$$ absolutely convergent, conditionally convergent, or divergent?
4. Is the series $$\sum_{i=6}^\infty(-1)^i\frac{i}{\sqrt{i^3-7}}$$ absolutely convergent, conditionally convergent, or divergent?
5. Is the series $$\sum_{m=2}^\infty(-\frac{3}{5})^m$$ absolutely convergent, conditionally convergent, or divergent?
6. Is the series $$\sum_{m=2}^\infty(-\frac{5}{3})^m$$ absolutely convergent, conditionally convergent, or divergent?
7. Is the series $$\sum_{n=13}^\infty(-1)^n\frac{1}{n\ln n}$$ absolutely convergent, conditionally convergent, or divergent?

Solutions

## 5.7 Ratio and Root Tests

• University Calculus: Early Transcendentals (3rd Ed)
• 9.5

### 5.7.1 Ratio Test

• The Ratio Test states that the series $$\sum_{n=N}^\infty a_n$$ absolutely converges when $$\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|<1$$ and diverges when $$\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|>1$$.
• Example Show that $$\sum_{n=0}^\infty\frac{3^n+1}{4^n}$$ absolutely converges using the Ratio Test. Then give its value.
• Example Does $$\sum_{k=3}^\infty\frac{(2k)!}{3(k!)^2}$$ converge or diverge?
• Another test must be used when $$\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|=1$$.
• Example Show that the divergent series $$\sum_{n=1}^\infty\frac{1}{n}$$ and the absolutely convergent series $$\sum_{n=1}^\infty\frac{1}{n^2}$$ both satisfy $$\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|=1$$.

### 5.7.2 Root Test

• The Root Test states that the series $$\sum_{n=N}^\infty a_n$$ absolutely converges when $$\lim_{n\to\infty}\sqrt[n]{|a_n|}<1$$ and diverges when $$\lim_{n\to\infty}\sqrt[n]{|a_n|}>1$$.
• Example Show that $$\sum_{n=0}^\infty\frac{5^n}{2^{3n}}$$ absolutely converges using the Root Test. Then give its value.
• Example Does $$\sum_{m=3}^\infty\frac{m^{10}}{(-3)^m}$$ converge or diverge?
• Another test must be used when $$\lim_{n\to\infty}\sqrt[n]{|a_n|}=1$$.

### Exercises for 5.7

1. Does $$\sum_{k=1}^\infty\frac{k^2+4}{(k+2)!}$$ converge or diverge?
2. Does $$\sum_{n=0}^\infty\frac{(2n)!}{n+3}$$ converge or diverge?
3. Does $$\sum_{m=2}^\infty\frac{5^m}{m!}$$ converge or diverge?
4. Does $$\sum_{n=0}^\infty(-1)^n\frac{n!}{2^n(n+2)!}$$ converge or diverge?
5. Does $$\sum_{p=0}^\infty\frac{3^p}{(p+7)^p}$$ converge or diverge?
6. Does $$\sum_{n=9}^\infty(1+\frac{2}{n})^{n^2}$$ converge or diverge? (Hint: $$e^x=\lim_{n\to\infty}(1+\frac{x}{n})^n$$.)
7. Does $$\sum_{j=3}^\infty(-3)^j\frac{1}{j4^j}$$ converge or diverge?
8. Does $$\sum_{n=1}^\infty\left(\frac{1-4n^2}{(n+1)(3n+1)}\right)^{n+3}$$ converge or diverge?
9. (OPTIONAL) Does $$\sum_{m=4}^\infty(-1)^{m+1}\frac{me^{-m}}{(2m+1)\ln(m+1)}$$ converge or diverge?
10. (QUIZ) Does $$\sum_{n=1}^\infty\frac{(n-1)!}{10^n}$$ converge or diverge?
• It converges.
• It diverges.
• It explodes.
11. (QUIZ) Does $$\sum_{k=3}^\infty(1-\frac{1}{k})^{k^2}$$ converge or diverge?
• It converges.
• It diverges.
• It converges some of the time, and diverges the rest of the time.
12. (QUIZ) Does $$\sum_{m=2}^\infty\frac{1}{m^2}$$ converge or diverge?
• It converges.
• It diverges.
• It is impossible to determine.

Solutions