Q. 3 What is meant by the sequence of... [FREE SOLUTION]

Chapter 7: Q. 3 (page 614)

What is meant by the sequence of partial sums for a series ${鈭�}_{k = 1}^{鈭�} a_{k}$ ? Why does it make sense to define the convergence of a series in terms of the convergence of its sequence of partial sums?

Short Answer

Expert verified

The sequence of partial sums of a series ${鈭�}_{k = 1}^{鈭�} a_{k}$ is the partial sum of the first through nth terms.

As $\{S_{n}\} = \{{鈭�}_{k = 1}^{n} a_{k}\} = \{a_{1}, (a_{1} + a_{2}), . . ., (a_{1} + a_{2} + . . . + a_{n})\}$

That's what, it makes a sense to define the convergence of a series in terms of the convergence of its sequence of partial sums.

Step by step solution

Step 1. Given information.

Consider the given question,

The series is ${鈭�}_{k = 1}^{鈭�} a_{k}$ .

Step 2. Explain the sequence of partial sums of the given series.

The sequence of partial sums of a series ${鈭�}_{k = 1}^{鈭�} a_{k}$ is the partial sum of the first through nth terms.

It can be defined as,

$S_{n} = a_{1} + a_{2} + . . . + a_{n}$

The sequence of partial sums is given by,

$\{S_{n}\} = \{{鈭�}_{k = 1}^{n} a_{k}\} = \{a_{1}, (a_{1} + a_{2}), . . ., (a_{1} + a_{2} + . . . + a_{n})\}$

That's what, it makes a sense to define the convergence of a series in terms of the convergence of its sequence of partial sums.

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91影视

What is meant by the sequence of partial sums for a series ${鈭�}_{k = 1}^{鈭�} a_{k}$ ? Why does it make sense to define the convergence of a series in terms of the convergence of its sequence of partial sums?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Explain the sequence of partial sums of the given series.

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91影视

What is meant by the sequence of partial sums for a series鈭�k=1鈭�ak? Why does it make sense to define the convergence of a series in terms of the convergence of its sequence of partial sums?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Explain the sequence of partial sums of the given series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Calculus

Probability and Statistics

Decision Maths

Theoretical and Mathematical Physics

Logic and Functions

Study anywhere. Anytime. Across all devices.

What is meant by the sequence of partial sums for a series ${鈭�}_{k = 1}^{鈭�} a_{k}$ ? Why does it make sense to define the convergence of a series in terms of the convergence of its sequence of partial sums?