   Chapter 11.8, Problem 34E

Chapter
Section
Textbook Problem

# Graph the first several partial sums sn(x) of the series ∑ n = 0 ∞ x n , together with the sum function f(x) = 1/(1 − x), on a common screen. On what interval do these partial sums appear to be converging to f(x)?

To determine

To sketch: The first several partial sums sn(x) of the series n=1xn ,together with the sum function f(x)=1(1x) and obtain the interval converging to f(x) .

Explanation

The given series is n=1xn

Let sn(x) be the nth partial sum of the series, sn=k=0nxk .

Here, an=xn .

The first five partial sum obtained as follows,

The first partial sum is, s0=a0 .

s0=x0=1

Thus, s0=1 .

The second partial sum is, s1=s0+a1 .

s1=s1+x1=1+x

Thus, s1=1+x .

The third partial sum is, s2=s1+a2 .

s2=s1+x2=1+x+x2

Thus, s2=1+x+x2 .

The fourth partial sum is, s3=s2+a3 .

s3=s2+a3=s2+x3=1+x+x2+x3

Thus, s3=1+x+x2+x3

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