Q.1 c) - Calculus

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

Step-by-Step Solution

Answer

false

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.