Find the interval of convergence for power series: ∑k=1∞1k(x+2)k

Short Answer

Expert verified

The interval of convergence for power series is[-3,-1).

Step by step solution

01

Step 1. Given information.  

The given power series is∑k=1∞1kx+2k.

02

Step 2. Find the interval of convergence. 

Let us assume bk=1kx+2k, therefore bk+1=1k+1x+2k+1

Ratio for the absolute convergence is

limk→∞bk+1bk=limk→∞1k+1x+2k+11kx+2k=limk→∞kx+2k+1=limk→∞x+2kk+1

Here, the limit is x+2. So, by the ratio test of absolute convergence. We know that the series will converge absolutely. Whenx+2<1that is-3<x<-1

03

Step 3. Find the interval of convergence. 

Now, since the intervals are finite so we analyse the behavior of the series at the endpoints.

So, when x=-3

∑k=1∞1kx+2k=∑k=1∞1k-3+2k=∑k=1∞1k-1k

The result is the alternating multiple of the harmonic series, which converges conditionally.

So, when x=-1

∑k=1∞1kx+2k=∑k=1∞1k-1+2k=∑k=1∞1k

The result is just a constant multiple of the harmonic series, which diverges.

Therefore, the interval of convergence of the power series is[-3,-1).

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