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Chapter 8: Q 21. (page 670)

Find the interval of convergence for power series:鈭�k=0鈭�1k!xk

Short Answer

Expert verified

The interval of convergence for power series is R.

Step by step solution

01

Step 1. Given information. 

The given power series is鈭�k=0鈭�1k!xk.

02

Step 2. Find the interval of convergence.

Let us assume bk=1k!xkandrole="math" localid="1649232298108" bk+1=1k+1!xk+1

Ratio for the absolute convergence is

limk鈫�鈭�bk+1bk=limk鈫�鈭�1k+1!xk+11k!xk=limk鈫�鈭�x1k+1

Now, we evaluate the limit at k鈫�鈭�.

So, limk鈫�鈭�x1k+1=0, that is the value of limit will be zero no matter what value the variable xtakes.

By ratio test, the series converges absolutely for every value of x.

Therefore, the interval of convergence of the power series鈭�k=0鈭�1k!xkis R.

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