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
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 \(r_n=\frac{n+2}{3+n^2}\)?
(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
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
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 \(\displaystyle\lim_{n\to\infty}\frac{\sin n + 3n^2}{n^2+1}\).
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)
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
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
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?
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
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?
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
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?
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
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
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
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?