Chapter 9.8, Problem 68E

### Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

Chapter
Section

### Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

# Investigation The interval of convergence of the series ∑ n − 0 ∞ ( 3 x ) n is   ( − 1 3 , 1 3 ) . (a) Find the sum of the series when x = 1 6 . Use a graphing utility to graph the First six terms of the sequence of partial sums and the horizontal line representing the sum of the series.(b) Repeat part (a) for x = − 1 6 (c) Write a short paragraph comparing the rates of convergence of the partial sums with the sums of the series in parts (a) and (b). How do the plots of the partial sums differ as they converge toward the sum of the series?(d) Given any positive real number M, there exists a positive integer N such that the partial sum ∑ n − 0 N ( 3 ⋅ 2 3 ) n > M . Use a graphing utility to complete the table. M 10 100 1000 10,000 N

(a)

To determine

To calculate: The sum of the series n=0(3x)n at the point x=16 when the geometric series converges on (13,13).

Explanation

Given: The interval of the convergence of the geometric series n=0(3x)n is the open interval (13,13).

Formula used: Geometric series

Calculation:

Consider the geometric series.

n=0(3x)n

And series converges on (13,13)

At, x=16, the series becomes

n=0(3x)n=n=0(316)n=n=0(12)n

The above represents a geometric series with a=0 and the ratio r=12.

Because the value of r lies in the interval 0|r|<1, the series converges and it’s sum is

s=a1r=1112=2

Graph the first ten terms of the sequence of partial sums with the help of graphing utility:

Step 1

(b)

To determine

To calculate: The sum of the series n=0(3x)n when x=16 as the geometric series converges on (13,13).

(c)

To determine
The comparison between the rates of convergence of the partial sums with the sums of the series in parts (a) and (b).

(d)

To determine

To calculate: The value of the positive integer N for given M.

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started