Absolute and Conditional Convergence

Theory

Absolute Convergence: A series $\sum a_n$ is absolutely convergent if $\sum |a_n|$ converges.
Conditional Convergence: A series $\sum a_n$ is conditionally convergent if $\sum a_n$ converges but $\sum |a_n|$ diverges.

Let $\sum a_n$ be an infinite series and suppose

L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|.

Exercise 1

Check whether the series converges:

\sum_{n=1}^\infty \frac{\cos(n\pi/3)}{n^2}

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Solution

Since $|\cos(n\pi/3)|\le 1$ ,

\left|\frac{\cos(n\pi/3)}{n^2}\right|\le \frac{1}{n^2}.

Because $\sum \frac{1}{n^2}$ converges, the given series converges absolutely.

Conclusion: Absolutely convergent.

Exercise 2

Check whether the series converges:

\sum_{n=1}^\infty \frac{(-5)^n}{n!}

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Solution

Apply the Ratio Test:

\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| =\lim_{n\to\infty}\frac{5}{n+1}=0&lt;1.

Conclusion: Absolutely convergent.

Exercise 3

Check whether the series converges:

\sum_{n=1}^\infty n\sin\left(\frac{1}{n}\right)

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Solution

Using the limit $\lim_{x\to 0}\frac{\sin x}{x}=1$ ,

\lim_{n\to\infty} n\sin\left(\frac{1}{n}\right)=1\neq 0.

By the Divergence Test, the series diverges.

Conclusion: Divergent.

Exercise 4

Check whether the series converges:

\sum_{n=1}^\infty \frac{(-1)^{n+1}}{\sqrt{n}}

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Solution

Let $a_n=\frac{1}{\sqrt{n}}$ . Then $a_n$ is decreasing and

\lim_{n\to\infty} a_n=0.

By the Alternating Series Test, the series converges.

However,

\sum_{n=1}^\infty \left|\frac{(-1)^{n+1}}{\sqrt{n}}\right| =\sum_{n=1}^\infty \frac{1}{\sqrt{n}}

diverges.

Conclusion: Conditionally convergent.