Chapter 5, Problem 379RE

True or False? Justify your answer with a proof or a counterexample. 379. If $\underset{n\to \infty }{\lim }{a}_n=0,$ then ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}{a}_n}$ converges.

To determine

To check: The given statement whether it is true or false and also justified it with a counterexample.

Answer to Problem 379RE

The statement is false.

Explanation of Solution

Given information:

The given statement is,

“If $\underset{n\to \infty }{\lim }{a}_n=0$ , then ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converges”.

Concept used:

By p-test,

The p-series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^p}}$ converges if $p>1$ and diverges if $p\le 1$ .

Calculation:

By Divergence Test,

If $\underset{n\to \infty }{\lim }{a}_n=c\ne 0$ or $\underset{n\to \infty }{\lim }{a}_n$ does not exist, then the series ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ diverges and if

  $\underset{n\to \infty }{\lim }{a}_n=0$ , the test is inconclusive.

Consider two series are:

  ${\displaystyle \sum _{n=1}^{\infty }{a}_n}={\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^3}}$

And,

  ${\displaystyle \sum _{n=1}^{\infty }{b}_n}={\displaystyle \sum _{n=1}^{\infty }\frac{1}{n}}$

Thus,

  $\begin{array}{c}\underset{n\to \infty }{\lim }{a}_n=\underset{n\to \infty }{\lim }\frac{1}{n^3}\\ =0\end{array}$

  $\begin{array}{c}\text{And, }\underset{n\to \infty }{\lim }{b}_n=\underset{n\to \infty }{\lim }\frac{1}{n}\\ =0\end{array}$

As the values of both $\underset{n\to \infty }{\lim }{a}_n$ and $\underset{n\to \infty }{\lim }{b}_n$ are equal to 0, their convergence and divergence will be computed by p-test.

By p-test,

The p-series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^p}}$ converges if $p>1$ and diverges if $p\le 1$ .

So, the series ${\displaystyle \sum _{n=1}^{\infty }{a}_n}={\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^3}}$ converges and the series ${\displaystyle \sum _{n=1}^{\infty }{b}_n}={\displaystyle \sum _{n=1}^{\infty }\frac{1}{n}}$ diverges.

Conclusion:

Hence, the given statement, “If $\underset{n\to \infty }{\lim }{a}_n=0$ , then ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converges” is false.