Question

if $0\le a_k\le \frac{1}{k}$for every positive integer k, then the series $\sum _{k=1}^{\infty }{a}_k$converges. The objective is to whether determine the statement is true or false

Short answer

false

Step-by-step solution

To determine whether given statement is true or false 

To determine whether the given statement is true or not by using comparison test

According to comparison test

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

If $\sum _{k=1}^{\infty }{b}_k$converges then the series $\sum _{k=1}^{\infty }{a}_k$also converges

If the series $\sum _{k=1}^{\infty }{b}_k=\sum _{k=1}^{\infty }\frac{1}{k}$is divergent by the power series, So $\sum _{k=1}^{\infty }{a}_k$also divergence

Hence the given statement is wrong.