Question

Explain why one convergence test can suffice to show that a series converges absolutely, even though it always requires two to show that a series converges conditionally.

Short answer

The series $\sum _{k=1}^{\infty }{a}_k$will be converge absolutely if $\sum _{k=1}^{\infty }\left|a_k\right|$converges. Hence, for Conditionally convergent series $\sum _{k=1}^{\infty }\left|a_k\right|$must diverges and $\sum _{k=1}^{\infty }\left|a_k\right|$converges . Hence for conditionally convergence the two test are required.

Step-by-step solution

Step 1. Given 

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

Step 2. Test for absolutely convergence

The series $\sum _{k=1}^{\infty }{a}_k$will be converge absolutely if $\sum _{k=1}^{\infty }\left|a_k\right|$converges. Hence, for absolutely convergent series it is sufficient to test that series$\sum _{k=1}^{\infty }\left|a_k\right|$ converges.

Step 3. Test for conditionally convergence 

The series $\sum _{k=1}^{\infty }{a}_k$will be converge absolutely if $\sum _{k=1}^{\infty }\left|a_k\right|$converges. Hence, for Conditionally convergent series $\sum _{k=1}^{\infty }\left|a_k\right|$must diverges and $\sum _{k=1}^{\infty }\left|a_k\right|$ converges . Hence for conditionally convergence the two test are required.