Question

What condition(s) must a series $\sum _{k=1}^{\infty }{a}_k$satisfy in order for the series to be absolutely convergent?

Short answer

$$\begin{gathered} \mathrm{The}\mathrm{series}\sum _{\mathrm{k}=1}^{\infty }{\mathrm{a}}_{\mathrm{k}}\mathrm{will}\mathrm{be}\mathrm{converge}\mathrm{absolutely}\mathrm{if}\sum _{\mathrm{k}=1}^{\infty }\left|{\mathrm{a}}_{\mathrm{k}}\right|\mathrm{converges}. \\ \mathrm{Hence},\mathrm{for}\mathrm{Conditionally}\mathrm{convergent}\mathrm{series}\sum _{\mathrm{k}=1}^{\infty }{\mathrm{a}}_{\mathrm{k}}\mathrm{must}\mathrm{converges}\mathrm{and}\sum _{\mathrm{k}=1}^{\infty }{\mathrm{a}}_{\mathrm{k}}\mathrm{converges}. \end{gathered}$$

Step-by-step solution

Step 1. Given

The given series is$\sum _{k=1}^{\infty }{a}_k$

Step 2. Test for absolutely convergence 

$$\begin{gathered} \mathrm{The}\mathrm{series}\sum _{\mathrm{k}=1}^{\infty }{\mathrm{a}}_{\mathrm{k}}\mathrm{will}\mathrm{be}\mathrm{converge}\mathrm{absolutely}\mathrm{if}\sum _{\mathrm{k}=1}^{\infty }\left|{\mathrm{a}}_{\mathrm{k}}\right|\mathrm{converges}. \\ \mathrm{Hence},\mathrm{for}\mathrm{Conditionally}\mathrm{convergent}\mathrm{series}\sum _{\mathrm{k}=1}^{\infty }{\mathrm{a}}_{\mathrm{k}}\mathrm{must}\mathrm{converges}\mathrm{and}\sum _{\mathrm{k}=1}^{\infty }{\mathrm{a}}_{\mathrm{k}}\mathrm{converges}. \end{gathered}$$