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Chapter 7: Problem 64

Define the binomial series. What is its radius of convergence?

Short Answer

Expert verified
The binomial series is represented as \((1+x)^k = 1 + kx + k(k-1)x^2/2! + k(k-1)(k-2)x^3/3! + ...\) for all x and for any real number k. The radius of convergence of the binomial series is 1.

Step by step solution

01

Define the binomial series

The binomial series is a sequence of numbers in which each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the ratio. It is represented as follows: \( (1+x)^k = 1 + kx + k(k-1)x^2/2! + k(k-1)(k-2)x^3/3! + ... \) for all x and for any real number k.
02

Calculate the ratio

To find the radius of convergence, we apply the ratio test, which states that if the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term of the series is less than 1, then the series converges. In the case of the binomial series, we find this limit is equal to \(|x|\), so the radius of convergence is 1.

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Most popular questions from this chapter

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \ln \left(\frac{n+1}{n}\right) $$

(a) find the common ratio of the geometric series, \((b)\) write the function that gives the sum of the series, and (c) use a graphing utility to graph the function and the partial sums \(S_{3}\) and \(S_{5} .\) What do you notice? $$ 1-\frac{x}{2}+\frac{x^{2}}{4}-\frac{x^{3}}{8}+\cdots $$

A fair coin is tossed repeatedly. The probability that the first head occurs on the \(n\) th toss is given by \(P(n)=\left(\frac{1}{2}\right)^{n},\) where \(n \geq 1\) (a) Show that \(\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^{n}=1\). (b) The expected number of tosses required until the first head occurs in the experiment is given by \(\sum_{n=1}^{\infty} n\left(\frac{1}{2}\right)^{n}\) Is this series geometric? (c) Use a computer algebra system to find the sum in part (b).

Find all values of \(x\) for which the series converges. For these values of \(x,\) write the sum of the series as a function of \(x\). $$ \sum_{n=0}^{\infty} 4\left(\frac{x-3}{4}\right)^{n} $$

Find the sum of the convergent series. $$ \sum_{n=1}^{\infty}(\sin 1)^{n} $$
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