Q. 11 - Calculus

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. 

Step-by-Step Solution

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 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.