Q. 9 - Calculus

Question

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

Step-by-Step Solution

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 1. Given

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

Step 2. Test for absolutely convergence

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