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Chapter 1: Q1. (page 29)

Use the ratio test to show that a binomial series converges for $| x | < 1$

Short Answer

Expert verified

The statement has been proven

Step by step solution

01

Given Information

The binomial series.

02

Definition of the binomial series

The Taylor series for the function given by is the binomial series, where is an arbitrary complex number.

03

Prove the statement.

The binomial series states that $(1 + x)^{p} = {鈭�}_{n = 0}^{鈭�} (\begin{array}{l} p \\ n \end{array}) x^{n}$

$(\begin{array}{l} p \\ n \end{array}) = \frac{p!}{n! (p - n)!} (1 + x)^{p}$

$= {鈭�}_{n = 0}^{鈭�} \frac{p!}{n! (p - n)!} X^{n} a_{n}$

$= \frac{p!}{n! (p - n)!} x^{n} a_{n + 1}$

$= \frac{p!}{(n + 1)! (p - n - 1)!} x^{n + 1}$

$Let 蚁_{n} = |\frac{a_{n + 1}}{a_{n}}|,$

Substitute the value of the power series in the formula above, the equation becomes as follows,

$蚁_{n} = |\frac{a_{n + 1}}{a_{n}}|$

$= |\frac{\frac{p!}{(n + 1)! (p - n - 1)!} x^{n + 1}}{\frac{p!}{n! n} x^{n}}|$

$= |\frac{p! n! x^{n + 1} (p - n)!}{(n + 1)! (p - n - 1)! p! x^{n}}|$

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Most popular questions from this chapter

By computer or tables, find the exact sum of each of the following series.

$(a) {鈭�}_{n = 1}^{鈭�} \frac{n}{{(n^{2} + 1)}^{2}} (b) {鈭�}_{n = 2}^{鈭�} \frac{In n}{n^{2}} (c) {鈭�}_{n = 1}^{鈭�} \frac{1}{n^{n}}$

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

$\lim_{x 鈫� 0} \frac{1 - \cos x}{x^{2}}$ .

$\ln (1 + x e^{x})$

In the following problems, find the limit of the given sequence asn $鈫� 鈭�$

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

$\lim_{x 鈫� 0} \frac{\sin x - x}{x^{3}}$ .

See all solutions

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