\( \newcommand{\<}{\langle} \newcommand{\>}{\rangle} \)

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

- 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 \(a_n=\frac{n}{2n+1}\).

- 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 \(a_0=3,a_{n+1}=\frac{a_n}{2}\).

- 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?

- 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 \(\<\frac{1}{3},-1,3,-9,27,\dots\>\).
- Create an explicit formula for each of the three previous sequences.
- 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 \(c_n=\frac{n}{1+n^2}\).
- Write the first six terms of the sequence \(\<q_n\>_{n=0}^\infty\) defined by \(q_0=0\) and \(q_{n+1}=q_n+2n+1\).
- Prove that \(q_n=n^2\) is an explicit formula for the sequence defined recursively in the previous problem.
- Write the first six terms of the sequence \(\<b_n\>_{n=1}^\infty\) defined by \(b_1=4\) and \(b_{n+1}=\frac{b_n}{2}\).
- Prove that \(b_n=\frac{8}{2^n}\) is an explicit formula for the sequence defined recursively in the previous problem.
- 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. - 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. - 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. - 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?
- 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?
- (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.
- (QUIZ)
What are the first five terms of the sequence \(\<r_n\>_{n=1}^\infty\)
defined explicitly by \(r_n=\frac{n+2}{3+n^2}\)?
- \(\<\frac{3}{4},\frac{4}{7},\frac{5}{12},\frac{6}{19}, \frac{1}{4},\dots\>\)
- \(\<\frac{2}{7},\frac{1}{2},\frac{4}{9},0,\frac{5}{17},\dots\>\)
- \(\<0,\frac{3}{5},\frac{5}{18},\frac{8}{27},\frac{9}{61},\dots\>\)

- (QUIZ)
What are the first five terms of the sequence \(\<w_n\>_{n=0}^\infty\)
defined recursively by \(w_0=1\), \(w_1=2\),
\(w_{n+2}=2w_n+w_{n+1}\)?
- \(\<1,2,5,10,17,\dots\>\)
- \(\<1,2,3,5,9,\dots\>\)
- \(\<1,2,4,8,16,\dots\>\)

- (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.

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

- 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 \(\lim_{n\to\infty}\frac{\sin n}{n}\).

- 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 \(\lim_{n\to\infty}(4n)^{1/n}\).

- 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.

- Use factoring to compute \(\displaystyle\lim_{n\to\infty}\frac{n-4n^2}{2n^2+7}\).
- Use L’Hopital’s Rule to prove that \(\displaystyle\frac{\ln n}{n}\to 0\).
- Use the squeeze theorem to compute \(\displaystyle\lim_{n\to\infty}\frac{\cos n}{n\ln n}\).
- Find \(\displaystyle\lim_{n\to\infty}\frac{\sin n + 3n^2}{n^2+1}\).
- Find \(\displaystyle\lim_{n\to\infty}\frac{\ln(n^n)}{n^2}\).
- Find \(\displaystyle\lim_{n\to\infty}(5n^3)^{2/n}\).
- Find \(\displaystyle\lim_{n\to\infty}(\frac{1}{\pi})^{3n}\).
- Find \(\displaystyle\lim_{n\to\infty}(\frac{1}{2}+\frac{1}{n})^n\).
- Find \(\displaystyle\lim_{n\to\infty} \frac{\frac{(n+2)!}{2^n}}{\frac{3n^2n!}{2^{n+1}}}\).
- 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?
- 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?
- 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?
- (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 \(L=e^x\).
- (QUIZ)
Find \(\lim_{n\to\infty}\frac{n!\cos n}{(n+1)!}\).
- \(1\)
- \(0\)
- \(\pi/2\)

- (QUIZ)
Find \(\lim_{n\to\infty}\frac{(3+n)^n}{n^n}\).
- \(1\)
- \(0\)
- \(e^3\)

- (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.

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

- 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.

- 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?

- 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.

- 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?

- Write out the first four terms of the partial sum sequence for \(\<1,-\frac{1}{3},\frac{1}{9},-\frac{1}{27},\dots\>\).
- Write out the first four terms of the partial sum sequence for \(\<0.3,0.03,0.003,0.0003,\dots\>\).
- Does \(\sum_{m=2}^\infty(\frac{3}{2m}-\frac{3}{2m+2})\) converge or diverge? If it converges, what is its value?
- Does \(\sum_{j=2}^\infty\frac{6}{4j^2+4j}\) converge or diverge? If it converges, what is its value?
- Compute \(1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\dots\).
- Prove that \(0.\overline3=0.333\dots\) equals \(\frac{1}{3}\) by expressing the decimal expression as a geometric series.
- Write \(0.\overline{27}=0.272727\dots\) as a fraction of integers.
- Does \(\sum_{n=0}^\infty\frac{6}{3^{n+2}}\) converge or diverge? If it converges, what is its value?
- Does \(\sum_{m=0}^\infty 3(-1)^m\) converge or diverge? If it converges, what is its value?
- Does \(\sum_{i=1}^\infty \frac{i+\sin i}{2i}\) converge or diverge? If it converges, what is its value?
- Suppose \(\sum_{n=0}^\infty a_n=3\) and \(\sum_{n=0}^\infty b_n=4\). Evaluate \(\sum_{n=0}^\infty(3a_n-2b_n)\).
- Does \(\sum_{k=2}^\infty 4(\frac{2}{3})^k\) converge or diverge? If it converges, what is its value?
- (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).
- (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.

- (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.

- (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.

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

- 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 \(\int_0^8 z^{-1/3}\,dz\).

- 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 \(\sum_{n=1}^\infty\frac{1}{n^3}\not=\int_1^\infty\frac{1}{x^3}\,dx\).

- 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?

- Does \(\int_2^\infty\frac{32}{x^3}\,dx\) converge or diverge? If it converges, what is its value?
- Does \(\int_0^\infty\frac{2y}{y^2+3}\,dy\) converge or diverge? If it converges, what is its value?
- Does \(\int_e^\infty\frac{1}{\ln(x^x)}\,dx\) converge or diverge? If it converges, what is its value?
- 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.
- Does \(\sum_{n=0}^\infty\frac{2n}{n^2+3}\) converge or diverge?
- Does \(\sum_{n=3}^\infty\frac{4}{n(\ln n)^3}\) converge or diverge?
- Does \(\sum_{n=-2}^\infty\frac{1}{e^n}\) converge or diverge?
- Show that \( \int_1^\infty\frac{1}{x^2}\,dx \not= \sum_{n=1}^\infty\frac{1}{n^2} \), even though they both converge.
- Does \(\sum_{k=100}^\infty\frac{5}{\sqrt[7]{k^6}}\) converge or diverge?
- Does \(\sum_{n=5}^\infty\frac{1}{n^2-8n+16}\) converge or diverge?
- (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 \(\lim_{u\to\infty}\tan^\leftarrow u=\frac{\pi}{2}\).)
- (QUIZ)
Does \(\sum_{m=0}^\infty\frac{2m}{(m^2+1)^2}\) converge or
diverge?
- It converges.
- It diverges.
- It both converges and diverges.

- (QUIZ)
Does \(\sum_{n=2}^\infty\frac{1}{\sqrt{n-1}}\) converge or
diverge?
- It converges.
- It diverges.
- It neither converges nor diverges.

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

- 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?

- 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?

- Does \(\sum_{n=0}^\infty\sqrt{\frac{n}{n^4+7}}\) converge or diverge? (Use Direct Comparison.)
- Does \(\sum_{n=3}^\infty\frac{4}{n^{0.8}-1}\) converge or diverge? (Use Direct Comparison.)
- Does \(\sum_{j=2}^\infty\frac{e^j}{e^{2j}+1}\) converge or diverge? (Use Direct Comparison.)
- Does \(\sum_{k=10}^\infty\frac{\sin^2(k)}{k^3}\) converge or diverge? (Use Direct Comparison.)
- Does \(\sum_{m=4}^\infty\frac{1}{\ln m}\) converge or diverge? (Use Direct Comparison.)
- Does \(\sum_{n=4}^\infty\frac{5}{2n+3}\) converge or diverge? (Use Direct Comparison.)
- Does \(\sum_{n=0}^\infty\sqrt{\frac{n}{n^4+7}}\) converge or diverge? (Use Limit Comparison.)
- Does \(\sum_{n=3}^\infty\frac{4}{n^{0.8}-1}\) converge or diverge? (Use Limit Comparison.)
- Does \(\sum_{j=2}^\infty\frac{e^j}{e^{2j}+1}\) converge or diverge? (Use Limit Comparison.)
- Does \(\sum_{k=10}^\infty\frac{\sin^2(k)}{k^3}\) converge or diverge? (Use Limit Comparison.)
- Does \(\sum_{m=4}^\infty\frac{1}{\ln m}\) converge or diverge? (Use Limit Comparison.)
- Does \(\sum_{n=4}^\infty\frac{5}{2n+3}\) converge or diverge? (Use Limit Comparison.)
- (OPTIONAL) Does \(\sum_{m=1}^\infty\frac{1}{1+2+\dots+(m-1)+m}\) converge or diverge? (Hint: show that \( \frac{1}{1+2+\dots+(m-1)+m} = \frac{2}{(1+m)+(2+m-1)+\dots+(m-1+2)+(m+1)} \).)
- (QUIZ)
Does \(\sum_{m=0}^\infty\frac{2m}{(m^2+1)^2}\) converge or
diverge?
- It converges.
- It diverges.
- It both converges and diverges.

- (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.

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

- 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.

- 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?

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

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

- 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 \(\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|=1\).

- 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 \(\lim_{n\to\infty}\sqrt[n]{|a_n|}=1\).

- Does \(\sum_{k=1}^\infty\frac{k^2+4}{(k+2)!}\) converge or diverge?
- Does \(\sum_{n=0}^\infty\frac{(2n)!}{n+3}\) converge or diverge?
- Does \(\sum_{m=2}^\infty\frac{5^m}{m!}\) converge or diverge?
- Does \(\sum_{n=0}^\infty(-1)^n\frac{n!}{2^n(n+2)!}\) converge or diverge?
- Does \(\sum_{p=0}^\infty\frac{3^p}{(p+7)^p}\) converge or diverge?
- Does \(\sum_{n=9}^\infty(1+\frac{2}{n})^{n^2}\) converge or diverge? (Hint: \(e^x=\lim_{n\to\infty}(1+\frac{x}{n})^n\).)
- Does \(\sum_{j=3}^\infty(-3)^j\frac{1}{j4^j}\) converge or diverge?
- Does \(\sum_{n=1}^\infty\left(\frac{1-4n^2}{(n+1)(3n+1)}\right)^{n+3}\) converge or diverge?
- (OPTIONAL) Does \(\sum_{m=4}^\infty(-1)^{m+1}\frac{me^{-m}}{(2m+1)\ln(m+1)}\) converge or diverge?
- (QUIZ)
Does \(\sum_{n=1}^\infty\frac{(n-1)!}{10^n}\) converge or
diverge?
- It converges.
- It diverges.
- It explodes.

- (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.

- (QUIZ)
Does \(\sum_{m=2}^\infty\frac{1}{m^2}\) converge or
diverge?
- It converges.
- It diverges.
- It is impossible to determine.