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Solved: Comparing remainder terms Use Exercise 73 to
Chapter 11, Problem 74AE(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. \(\sum_{k=0}^{\infty} 0.6^{k}\) b. \(\sum_{k=0}^{\infty} 0.15^{k}\)
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. \(\sum_{k=0}^{\infty} 0.6^{k}\) b. \(\sum_{k=0}^{\infty} 0.15^{k}\)
ANSWER:Step 1 of 5
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 =
=
, provide if |r| < 1……..(1)
and let =
=
, be the sum of first n terms ……….(2)
The remainder is the error approximating S by
.
Now , we have to show that = |S -
| = |
|
From ( 1) , and (2)
= |S -
| = |
-
|
= ||, since denominators are equal.
= ||
Therefore , = |S -
| = |
|.............(3)