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Find all values of x for which the series k=1x2kk2converges.

Short Answer

Expert verified

Seriesk=1x2kk2converses whenx(-1,1).

Step by step solution

01

Step 1. Given information.

The given series isk=1x2kk2.

02

Step 2. Value of x.

Use the ratio test and Find the value of ρ=limkak+1ak

role="math" localid="1649196887689" ρ=limkx2(k+1)k+12x2kk2=limkx2k·x2·k2k+12·x2k=x2·limkkk+12=x2

According to the ratio test, series will converge when ρ<1.

role="math" localid="1649197012614" x2<1-1<x<1

So series converses whenrole="math" localid="1649197120658" x(-1,1)

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