Question

If$\frac{1}{k^2}<b_k$for every positive integer k, then the series $\sum _{k=1}^{\infty }{b}_k$diverges. The objective is to determine whether statement is true or false.

Short answer

False

Step-by-step solution

To determine whether given statement is true or false

To determine whether given statement is true or false by comparison test

According to comparison test

$\sum _{k=1}^{\infty }{a}_k$and$\sum _{k=1}^{\infty }{b}_k$be the two series with the positive terms such that $0\le a_k\le b_k$for every positive integer k

If the series $\sum _{k=1}^{\infty }{b}_k$is convergence , then the series $\underset{k=1}{\overset{\infty }{\sum a_k}}$also converges.

we cannot able to determine the convergence and divergence of$\sum _{k=1}^{\infty }{b}_k$

we cannot say about the behavior of the series $\sum _{k=1}^{\infty }{b}_k$if $\frac{1}{k^2}<b_{k_{}}$holds .

Hence the given statement is false .