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Chapter 8: Q 45. (page 670)

Find the interval of convergence for power series:鈭�k=1鈭�12.4.6.......2kxk

Short Answer

Expert verified

The interval of convergence for power series isR.

Step by step solution

01

Step 1. Given information. 

The given power series is鈭�k=1鈭�12.4.6.......2kxk.

02

Step 2. Find the interval of convergence.  

Let us assume bk=12.4.6......2kxktherefore bk+1=12.4.6.......2k+1xk+1

Ratio for the absolute convergence is

limk鈫�鈭�bk+1bk=limk鈫�鈭�12.4.6......2k+1xk+112.4.6.....2kxk=limk鈫�鈭�12k+1x

Since, for k鈫�鈭�, the limit is zero irrespective of the value of variable.

This implies that

limk鈫�鈭�12k+1x=0

Hence by the ratio test the series converges absolutely for every value of x.

Therefore, the interval of convergence of the power series isR.

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