Question

Prove that if the series $\sum _{k=1}^{\mathrm{\infty }} a_k$diverges, then the series $\sum _{k=1}^{\mathrm{\infty }} \left|a_k\right|$also diverges.

Short answer

The series $\sum _{k=1}^{\mathrm{\infty }} a_k$is convergent

The series$\sum _{k=1}^{\mathrm{\infty }} \left|a_k\right|$is divergent

Step-by-step solution

Step 1. Given information

$\sum _{k=1}^{\mathrm{\infty }} \left|a_k\right|$and$\sum _{k=1}^{\mathrm{\infty }} a_k$are the given series

Step 2. Finding whether ∑k=1∞ akis convergent

Assume that $\sum _{k=1}^{\mathrm{\infty }} \left|a_k\right|$is not divergent,

Therefore, $\sum _{k=1}^{\mathrm{\infty }} \left|a_k\right|$is convergent.

If $\sum _{k=1}^{\mathrm{\infty }} a_k$is convergent but it is given that it is divergent.

If the series $\sum _{k=1}{a}_k$is absolutely convergent, then the series $\sum _{k=1}{a}_k$is convergent.

Therefore, $\sum _{k=1}^{\infty }{a}_k$is convergent, which is a contradiction as it is given that the series $\sum _{k=1}^{\infty }{a}_k$is divergent.

Therefore, the supposition that the series $\sum _{k=1}^{\mathrm{\infty }} \left|a_k\right|$is not divergent is wrong.

Hence, the series$\sum _{k=1}^{\mathrm{\infty }} \left|a_k\right|$is divergent.