Q. 12 - Calculus

Question

 Explain why you must use two convergence tests to show that a series $\sum _{k=1}^{\infty }{a}_{\mathrm{k}} \mathrm{Converges} \mathrm{conditionally}$

Step-by-Step Solution

Answer

$\mathrm{For} \mathrm{conditionally} \mathrm{convergent} \mathrm{series} \underset{\mathrm{k}=1}{\overset{\infty }{\sum }}\left|{\mathrm{a}}_{\mathrm{k}}\right| \mathrm{must} \mathrm{diverge} \mathrm{and} \sum _{\mathrm{k}=1}^{\infty }{\mathrm{a}}_{\mathrm{k}} \mathrm{converges} .$

Step 1. Given

$\sum _{k=1}^{\infty }{a}_{\mathrm{k}} \mathrm{Converges} \mathrm{conditionally}$.

Step 2. Explanation

$$\begin{gathered} \mathrm{The} \mathrm{series} \sum _{\mathrm{k}=1}^{\infty }{\mathrm{a}}_{\mathrm{k}} \mathrm{will} \mathrm{be} \mathrm{converge} \mathrm{conditionally} \mathrm{if} \underset{\mathrm{k}=1}{\overset{\infty }{\sum }}\left|{\mathrm{a}}_{\mathrm{k}}\right| \mathrm{Diverges} . \\ \mathrm{Hence} , \mathrm{for} \mathrm{conditionally} \mathrm{convergent} \mathrm{series} \underset{\mathrm{k}=1}{\overset{\infty }{\sum }}\left|{\mathrm{a}}_{\mathrm{k}}\right| \mathrm{must} \mathrm{diverge} \mathrm{and} \sum _{\mathrm{k}=1}^{\infty }{\mathrm{a}}_{\mathrm{k}} \mathrm{converges} . \\ \mathrm{Hence} , \mathrm{for} \mathrm{conditionally} \mathrm{convergent} \mathrm{series} \mathrm{two} \mathrm{test} \mathrm{are} \mathrm{required}. \end{gathered}$$