Q. 10 Give precise mathematical defini... [FREE SOLUTION]

Chapter 7: Q. 10 (page 655)

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition or description with a graph or an algebraic example.
The remainder of a convergent series

Short Answer

Expert verified

The difference between the nth partial sum and the sum of a series is the remainder of the convergent series.

For example, in the sequence, $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + . . .$ the remainder of the convergent series is $R_{1} = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + . . . ., R_{2} = \frac{1}{8} + \frac{1}{16},,,, R_{n} = \frac{1}{2^{n + 1}} + \frac{1}{2^{n + 2}} + \frac{1}{2^{n + 3}} + . . .$

Step by step solution

Step 1. Given Information

The given statement is the remainder of a convergent series

Step 2. Explanation

The difference between the nth partial sum and the sum of a series is the remainder of the convergent series.

For example, consider a sequence $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + . .$

The first few partial sums and remainders are as follows,

$S_{1} = \frac{1}{2}, R_{1} = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + . . . S_{2} = \frac{1}{2} + \frac{1}{4}, R_{2} = \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + . . .$

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Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition or description with a graph or an algebraic example.
The remainder of a convergent series

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

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91影视

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition or description with a graph or an algebraic example. The remainder of a convergent series

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

Pure Maths

Calculus

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition or description with a graph or an algebraic example.
The remainder of a convergent series