Chapter 1: Q-24E (page 1)

Question: In Problems 23–26, express the given power series as a series
with generic term X^k.
24. $\sum_{n = 2}^{\infty} n (n - 1) a_{n} x^{n + 2}$

Short Answer

Expert verified

The required term is . $\sum_{k = 4}^{\infty} (k - 2) (k - 3) a_{k - 2} x^{k}$

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, represents the coefficient term of the nth term, 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, $f (x) = \sum_{n = 2}^{\infty} n (n - 1) a_{n} x^{n + 2}$ .

Let,

$n + 2 = k n = k - 2$

So,

$\sum_{n = 2}^{\infty} n (n - 1) a_{n} x^{n + 2} = \sum_{k - 2 = 2}^{\infty} (k - 2) (k - 2 - 1) a_{k - 2} x^{k} \sum_{n = 2}^{\infty} n (n - 1) a_{n} x^{n + 2} = \sum_{k = 4}^{\infty} (k - 2) (k - 2 - 1) a_{k - 2} x^{k}$

Hence, the required term is . $\sum_{k = 4}^{\infty} (k - 2) (k - 2 - 1) a_{k - 2} x^{k}$ .

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Question: In Problems 23–26, express the given power series as a series
with generic term X^k.
24. $\sum_{n = 2}^{\infty} n (n - 1) a_{n} x^{n + 2}$

Short Answer

Step by step solution

Power series

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

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Question: In Problems 23–26, express the given power series as a serieswith generic term Xk.24.∑n=2∞n(n-1)anxn+2

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

Logic and Functions

Geometry

Theoretical and Mathematical Physics

Applied Mathematics

Probability and Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Question: In Problems 23–26, express the given power series as a series
with generic term X^k.
24. $\sum_{n = 2}^{\infty} n (n - 1) a_{n} x^{n + 2}$