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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

Expert verified

The series k=1akwill be converge absolutely if k=1akconverges. Hence, for Conditionally convergent series k=1akmust diverges and k=1akconverges . Hence for conditionally convergence the two test are required.

Step by step solution

01

Step 1. Given 

The given series isk=1ak

02

Step 2. Test for absolutely convergence

The series k=1akwill be converge absolutely if k=1akconverges. Hence, for absolutely convergent series it is sufficient to test that seriesk=1ak converges.

03

Step 3. Test for conditionally convergence 

The series k=1akwill be converge absolutely if k=1akconverges. Hence, for Conditionally convergent series k=1akmust diverges and k=1ak converges . Hence for conditionally convergence the two test are required.

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