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Chapter 1: Q6 (page 22)

鈭�n=1鈭�(-1)nxn(2n)!

Short Answer

Expert verified

Hence the given power series is convergent for all the values ofx

Step by step solution

01

Given Information 

The power series is 鈭�n=1鈭�(-1)nxn(2n)!

02

Definition of the interval of convergence.

The Interval of convergence is the interval in which the power series is convergent.

03

Find the interval.

The power series is鈭�n=1鈭�(-1)nxn(2n)!

Let 蚁n=|an+1an|.

Substitute the value of the power series in the formula above, the equation becomes as follows.

蚁n=|an+1an|

=|xn+1(2n+2)!xn(2n)!|=|xn+1(2n)!(2n+2)!xn|

Apply limits in the above equation.

蚁=limn鈫�鈭�|x(2n+2)(2n+1)|=|x鈭�|=0

The power series is convergent for 蚁<1 .

Hence the given power series is convergent for all the values of x

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