3.3 Convergence Tests

Select the series below that diverge by the divergence test.

(A)  $\displaystyle\sum_{n=1}^\infty \frac{1}{n}$ (B)  $\displaystyle\sum_{n=1}^\infty \frac{n^2}{n+1}$ (C) $\displaystyle\sum_{n=1}^\infty \sin n$ (D)  $\displaystyle\sum_{n=1}^\infty \sin (\pi n)$

Select the series below whose terms satisfy the conditions to apply the integral test.

(A)  $\displaystyle\sum_{n=1}^\infty \frac{1}{n}$ (B)  $\displaystyle\sum_{n=1}^\infty \frac{n^2}{n+1}$ (C) $\displaystyle\sum_{n=1}^\infty \sin n$ (D)  $\displaystyle\sum_{n=1}^\infty \frac{\sin n+1}{n^2}$

Suppose there is some threshold after which a person is considered old, and before which they are young.

Let Olaf be an old person, and let Yuan be a young person.

Below are graphs of two sequences with positive terms. Assume the sequences continue as shown. Fill in the table with conclusions that can be made from the direct comparison test, if any.

For each pair of series below, decide whether the second series is a valid comparison series to determine the convergence of the first series, using the direct comparison test and/or the limit comparison test.

Suppose $a_n$ is a sequence with $\displaystyle\lim_{n \to \infty}a_n = \frac{1}{2}$. Does $\displaystyle\sum_{n=7}^\infty a_n$ converge or diverge, or is it not possible to determine this from the information given? Why?

What flaw renders the following reasoning invalid?

Q: Determine whether $\displaystyle\sum_{n=1}^\infty \dfrac{\sin n}{n}$ converges or diverges.
A: First, we will evaluate $\displaystyle\lim_{n \to \infty} \dfrac{\sin n}{n}$.

So, by the divergence test, $\displaystyle\sum_{n=1}^\infty \dfrac{\sin n}{n}$ converges.

What flaw renders the following reasoning invalid?

Q: Determine whether $\displaystyle\sum_{n=1}^\infty \left(\sin(\pi n)+2\right)$ converges or diverges.
A: We use the integral test. Let $f(x)=\sin(\pi x)+2$. Note $f(x)$ is always positive, since $\sin(x)+2 \geq -1+2 =1$. Also, $f(x)$ is continuous.

By the integral test, since the integral diverges, also $\displaystyle\sum_{n=1}^\infty\left( \sin(\pi n)+2\right)$ diverges.

What flaw renders the following reasoning invalid?

Q: Determine whether the series $\displaystyle\sum_{n=1}^\infty \dfrac{2^{n+1}n^2}{e^n+2n}$ converges or diverges.
A: We want to compare this series to the series $\displaystyle\sum_{n=1}^\infty \dfrac{2^{n+1}}{e^n}$. Note both this series and the series in the question have positive terms.

First, we find that $\dfrac{2^{n+1}n^2}{e^n+2n} > \dfrac{2^{n+1}}{e^n}$ when $n$ is sufficiently large. The justification for this claim is as follows:

Now, we claim $\displaystyle\sum_{n=1}^\infty \dfrac{2^{n+1}}{e^n}$ converges.
Note $\displaystyle\sum_{n=1}^\infty \dfrac{2^{n+1}}{e^n}= 2\displaystyle\sum_{n=1}^\infty \dfrac{2^{n}}{e^n}= 2\displaystyle\sum_{n=1}^\infty \left(\dfrac{2}{e}\right)^n$. This is a geometric series with $r=\frac{2}{e}$. Since $2/e <1$, the series converges.

Now, by the Direct Comparison Test, we conclude that $\displaystyle\sum_{n=1}^\infty \dfrac{2^{n+1}n^2}{e^n+2n}$ converges.

Which of the series below are alternating?

(A) $\displaystyle\sum_{n=1}^\infty \sin n$ (B) $\displaystyle\sum_{n=1}^\infty \frac{\cos(\pi n)}{n^3}$ (C) $\displaystyle\sum_{n=1}^\infty \frac{7}{(-n)^{2n}}$ (D) $\displaystyle\sum_{n=1}^\infty \frac{(-2)^n}{3^{n+1}}$

Give an example of a convergent series for which the ratio test is inconclusive.

Imagine you're taking an exam, and you momentarily forget exactly how the inequality in the ratio test works. You remember there's a ratio, but you don't remember which term goes on top; you remember there's something about the limit being greater than or less than one, but you don't remember which way implies convergence.

Explain why

or, equivalently,

should mean that the sum $\sum\limits_{n=1}^\infty a_n$ diverges (rather than converging).

Give an example of a series $\displaystyle\sum_{n=a}^\infty a_n$, with a function $f(x)$ such that $f(n)=a_n$ for all whole numbers $n$, such that:

Suppose that you want to use the Limit Comparison Test on the series $\displaystyle \sum_{n=0}^{\infty} a_n$ where $\displaystyle a_n = \frac{2^n+n}{3^n+1}$. Write down a sequence $\{b_n\}$ such that $\displaystyle \lim\limits_{n\to\infty} \frac{a_n}{b_n}$ exists and is nonzero. (You don't have to carry out the Limit Comparison Test)

Decide whether each of the following statements is true or false. If false, provide a counterexample. If true provide a brief justification.

Does the series $\displaystyle \sum_{n=2}^\infty \frac{n^2}{3n^2+\sqrt n}$ converge?

Determine, with explanation, whether the series $\displaystyle \sum_{n=1}^\infty \frac{5^k}{4^k+3^k}$ converges or diverges.

Determine whether the series $\displaystyle\sum_{n=0}^\infty\frac{1}{n+\frac{1}{2}}$ is convergent or divergent. If it is convergent, find its value.

Does the following series converge or diverge? $\displaystyle\sum_{k=1}^\infty\frac{1}{\sqrt{k}\sqrt{k+1}}$

Evaluate the following series, or show that it diverges: $\displaystyle\sum_{k=30}^\infty 3(1.001)^k$.

Evaluate the following series, or show that it diverges: $\displaystyle\sum_{n=3}^\infty \left(\frac{-1}{5}\right)^n$.

Does the following series converge or diverge? $\displaystyle\sum_{n=7}^\infty \sin(\pi n)$

Does the following series converge or diverge? $\displaystyle\sum_{n=7}^\infty\cos(\pi n)$

Does the following series converge or diverge? $\displaystyle\sum_{k=1}^\infty \frac{e^k}{k!}$.

Evaluate the following series, or show that it diverges: $\displaystyle\sum_{k=0}^\infty\frac{2^k}{3^{k+2}}$.

Does the following series converge or diverge? $\displaystyle\sum_{n=1}^\infty\frac{n!n!}{(2n)!}$.

Does the following series converge or diverge? $\displaystyle\sum_{n=1}^\infty\frac{n^2+1}{2n^4+n}$.

Show that the series $\displaystyle\sum_{n=3}^\infty \frac{5}{n(\log n)^{3/2}}$ converges.

Find the values of $p$ for which the series $\displaystyle{\sum_{n=2}^\infty \frac{1}{n(\log n)^p}}$ converges.

Does ${\displaystyle\sum_{n=1}^\infty\frac{e^{-\sqrt{n}}}{\sqrt{n}}}$ converge or diverge?

Use the comparison test (not the limit comparison test) to show whether the series
$\displaystyle \sum_{n=2}^{\infty} \frac{\sqrt{3 n^2 - 7}}{n^{3}}$ converges or diverges.

Determine whether the series $\displaystyle\sum_{k=1}^\infty\frac{ \sqrt[3]{k^4+1} } {\sqrt{k^5+9}}$ converges.

Does $\displaystyle\sum_{n=1}^\infty\frac{n^4 2^{n/3}}{(2n+7)^4}$ converge or diverge?

Determine, with explanation, whether each of the following series converge or diverge.

Determine whether the series

converges or diverges.

Determine whether each of the following series converge or diverge.

Evaluate the following series, or show that it diverges: $\displaystyle\sum_{n=5}^\infty \frac{1}{e^n}$.

Determine whether the series $\displaystyle\sum_{n=2}^\infty\frac{6}{7^n}$ is convergent or divergent. If it is convergent, find its value.

Determine, with explanation, whether each of the following series converge or diverge.

Determine, with explanation, whether each of the following series converges or diverges.

Determine whether the series $\displaystyle\sum_{n=1}^\infty\frac{n^3-4}{2n^5-6n}$ is convergent or divergent.

What is the smallest value of $N$ such that the partial sum $\displaystyle\sum_{n=1}^N\frac{(-1)^n}{n\cdot 10^n}$ approximates $\displaystyle\sum_{n=1}^\infty\frac{(-1)^n}{n\cdot 10^n}$ within an accuracy of $10^{-6}$?

It is known that $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2} = \frac{\pi^2}{12}$ (you don't have to show this). Find $N$ so that $S_N$, the $N^{\rm th}$ partial sum of the series, satisfies $| \frac{\pi^2}{12} - S_N | \le 10^{-6}$. Be sure to say why your method can be applied to this particular series.

The series $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n+1}}{(2n+1)^2}$ converges to some number $S$ (you don't have to prove this). According to the Alternating Series Estimation Theorem, what is the smallest value of $N$ for which the $N^{\rm th}$ partial sum of the series is at most $\frac1{100}$ away from $S$? For this value of $N$, write out the $N^{\rm th}$ partial sum of the series.

Determine, with explanation, whether the following series converge or diverge.

(a) Prove that $\displaystyle \int_2^\infty\frac{x+\sin x}{1+x^2}\ \dee{x}$ diverges.

(b) Explain why you cannot conclude that $\displaystyle\sum\limits_{n=1}^\infty \frac{n+\sin n}{1+n^2}$ diverges from part (a) and the Integral Test.

(c) Determine, with explanation, whether $\displaystyle\sum\limits_{n=1}^\infty \frac{n+\sin n}{1+n^2}$ converges or diverges.

Show that $\displaystyle\sum\limits_{n=1}^\infty\frac{e^{-\sqrt{n}}}{\sqrt{n}}$ converges and find an interval of length $0.05$ or less that contains its exact value.

Suppose that the series $\displaystyle\sum\limits_{n=1}^\infty a_n$ converges and that $1>a_n\ge 0$ for all $n$. Prove that the series $\displaystyle\sum\limits_{n=1}^\infty \frac{a_n}{1-a_n}$ also converges.

Suppose that the series $\sum\limits_{n=0}^{\infty}(1-a_n)$ converges, where $a_n>0$ for $n=0,1,2,3,\cdots$. Determine whether the series $\sum\limits_{n=0}^\infty 2^n a_n$ converges or diverges.

Assume that the series $\displaystyle\sum_{n=1}^\infty\frac{na_n-2n+1}{n+1}$ converges, where $a_n > 0$ for $n = 1, 2, \cdots$. Is the following series

convergent? If your answer is NO, justify your answer. If your answer is YES, evaluate the sum of the series $-\log a_1 + \sum\limits_{n=1}^\infty \log\big(\frac{a_n}{a_{n+1}}\big)$.

Prove that if $a_n\ge 0$ for all $n$ and if the series $\displaystyle\sum_{n=1}^\infty a_n$ converges, then the series $\displaystyle\sum_{n=1}^\infty a^2_n$ also converges.