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Prove that if k=1akis a convergent series with ak0for every positive integer k, then the series k=1akkconverges.

Short Answer

Expert verified

Series k=1akis a convergent where ak0for every positive integer k.

so N>0will also exist such that ak<1for alln>N.

when n>N&ak<1,then0akkak.

According to The Comparison Test, if role="math" localid="1649139323101" 0akkak&k=1akconverges then series k=1akkalso converges.

Step by step solution

01

Step 1. Given Information. 

The given series isk=1akk.

Series k=1akis a convergent where ak1for every positive integerk.

02

Step 2. Proof. 

Series k=1akis a convergent where ak0for every positive integer k.

so N>0will also exist such that ak<1for alln>N.

when n>N&ak<1,thenrole="math" localid="1649139290641" 0akkak.

According to The Comparison Test, if role="math" localid="1649139270963" 0akkak&k=1akconverges then series k=1akkalso converges.

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