Solution Found!
Answer: Comparing remainder terms Use Exercise 73 to
Chapter 11, Problem 77AE(choose chapter or problem)
74-77. Comparing remainder terms Use Exercise 73 to determine how many terms of each series are needed so that the partial sum is within \(10^{-6}\) of the value of the series (that is, to ensure \(R_{n}<10^{-6}\)).
a. \(\sum_{k=0}^{\infty}\left(\frac{1}{\pi}\right)^{k}\) b. \(\sum_{k=0}^{\infty}\left(\frac{1}{e}\right)^{k}\)
Questions & Answers
QUESTION:
74-77. Comparing remainder terms Use Exercise 73 to determine how many terms of each series are needed so that the partial sum is within \(10^{-6}\) of the value of the series (that is, to ensure \(R_{n}<10^{-6}\)).
a. \(\sum_{k=0}^{\infty}\left(\frac{1}{\pi}\right)^{k}\) b. \(\sum_{k=0}^{\infty}\left(\frac{1}{e}\right)^{k}\)
ANSWER:Problem 77AE
Comparing remainder terms Use Exercise 73 to determine how many terms of each series are needed so that the partial sum is within of the value of the series (that is, to ensure
).
Solution
Step 1
In this problem we need to find many terms of each series are needed so that the partial sum is within of the value of the series.
That is we have to find the value of n so that the remainder in finding the sum of series from 0 to and the sum of series from
to n is less than
First let us see how to find the remainder term.
Remainder term Consider the geometric series , which has the value
provided |r|<1, and let
be the sum of the first n terms. The remainder
is the error in approximating S by Sn which is given by
… (1)