Chapter 1: Q-25E (page 1)

Question: In Problems 23–26, express the given power series as a series with generic term X^K.
25. $\sum_{n = 0}^{\infty} a_{n} x^{n + 1}$

Short Answer

Expert verified

The required term is $\sum_{k = 1}^{\infty} a_{k - 1} x^{k} .$

Step by step solution

Power series

A power series is an infinite series of the form,

$\sum_{n = 0}^{\infty} a_{n} {(x - c)}^{n} = a_{0} + a_{1} (x - c) + a_{2} (x - c)^{2} + . . . . .$

Where,a_n represents the coefficient term of the nth term and c is a constant.

To express the given series in terms of a generic term xk

In order to express the given series in terms of generic term x^k , we will change the index of the power series.

Given that, $f (x) = \sum_{n = 0}^{\infty} a_{n} x^{n + 1}$

Let,

$n + 1 = k n = k - 1$

So,

. $\sum_{n = 0}^{\infty} a_{n} x^{n + 1} = \sum_{k - 1 = 0}^{\infty} a_{k - 1} x^{k} \sum_{n = 0}^{\infty} a_{n} x^{n + 1} = \sum_{k = 1}^{\infty} a_{k - 1} x^{k}$

Hence, the required term is $\sum_{k = 1}^{\infty} a_{k - 1} x^{k}$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Question: In Problems 23–26, express the given power series as a series with generic term X^K.
25. $\sum_{n = 0}^{\infty} a_{n} x^{n + 1}$

Short Answer

Step by step solution

Power series

To express the given series in terms of a generic term xk

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Geometry

Applied Mathematics

Decision Maths

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

Question: In Problems 23–26, express the given power series as a series with generic term XK.25.∑n=0∞anxn+1

Short Answer

Step by step solution

Power series

To express the given series in terms of a generic term xk

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Geometry

Applied Mathematics

Decision Maths

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

Question: In Problems 23–26, express the given power series as a series with generic term X^K.
25. $\sum_{n = 0}^{\infty} a_{n} x^{n + 1}$