Chapter 11.2, Problem 4PQ

To determine

A student asserts that ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2}}=0$ because $\frac{1}{n^2}$ tends to zero. Is this valid reasoning.

No, this is not valid reasoning because thhe partial sums of the given series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2}}$ are

$\begin{array}{l}{s}_1=\frac{1}{1}\\ s_2=1+\frac{1}{4}=\frac{5}{4}\\ s_3=1+\frac{1}{4}+\frac{1}{9}=\frac{49}{36}\end{array}$

So, ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2}}\ne 0$

Given information:

${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2}}\ne 0$

Formula used:

In fact the nth term tending to zero for large value of $n$ does not imply that the sum of the series in zero.

Calculation:

No, this is not valid reasoning because the partial sums of the given series. ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2}}$ are

$\begin{array}{l}{s}_1=\frac{1}{1}\\ s_2=1+\frac{1}{4}=\frac{5}{4}\\ s_3=1+\frac{1}{4}+\frac{1}{9}=\frac{49}{36}\end{array}$

So, ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2}}\ne 0$

In fact the nth term tending to zero for large value of $n$ does not imply that the sum of the series in zero.