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Chapter 11: Problem 9

Exer. 1-12: Find a power series representation for the expression, and state the radius of convergence. $$ (1+x)^{-3} $$

Short Answer

Expert verified
The series is \(\sum_{n=0}^{\infty} \frac{(n+2)(n+1)}{2} (-1)^n x^n\), with radius of convergence 1.

Step by step solution

01

Identify the Basic Function Form

Recognize that the expression \((1 + x)^{-3}\) can be related to the standard geometric series formula. Recollect that the geometric series \(\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n\), valid for \(|x| < 1\), provides the base strategy for power series representation.
02

Determine the Series for \((1+x)^{-1}\)

Since we have \((1+x)^{-3}\), note that it can be seen as the derivative of a simpler form related to \((1-x)^{-1}\). First, rewrite \((1+x)^{-1}\) by considering its similarity to \(\frac{1}{1-(-x)} = \sum_{n=0}^{\infty} (-x)^n = \sum_{n=0}^{\infty} (-1)^n x^n\).
03

Differentiate for Exponent 3

To find the expression for \((1+x)^{-3}\), recognize the general pattern for such functions: \((1+x)^{-k} = \sum_{n=0}^{\infty} \binom{k+n-1}{n} (-1)^n x^n\) for integer \(k\). Here \(k = 3\), so substitute to get the series: \(\sum_{n=0}^{\infty} \binom{2+n}{n} (-1)^n x^n\).
04

Calculate Binomial Coefficients

Compute \(\binom{2+n}{n}\) which is generally computed as \(\frac{(2+n)!}{2!n!}\). Calculate it to express the coefficient more explicitly: \(\binom{2+n}{n} = \frac{(2+n)(1+n)}{2}\).
05

Write the Series Representation

Thus the power series for \((1+x)^{-3}\) is given as \(\sum_{n=0}^{\infty} \frac{(n+2)(n+1)}{2} (-1)^n x^n\). Simplify if needed.
06

Determine Radius of Convergence

Observe that the series polynomial form related to \((1+x)^n\) has an interval of convergence \(|x|<1\). Therefore, applying this understanding, the original function \((1+x)^{-3}\) maintains a radius of convergence of 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radius of Convergence
A power series has a radius of convergence, which is a critical part of understanding its properties. The radius of convergence tells us the span of values where the series converges to a particular function.
For the expression \((1+x)^{-3}\), the radius of convergence is determined using standard results from geometric series.
  • The geometric series \(\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n\) converges for \(|x| < 1\).
  • Similarly, the power series for \((1+x)^{-3}\) retains this interval, indicating that it converges when \(|x| < 1\).
Thus, we say that the radius of convergence for this power series is 1. This means that the series provides a valid representation of the function for values of \(x\) within this interval, ensuring our calculations and expectations are consistent.
Geometric Series
Geometric series are fundamental when discussing power series and convergence.
A geometric series is a series of the form \(\sum_{n=0}^{\infty} ar^n\), where \(a\) is the first term and \(r\) is the common ratio.For a basic geometric series, when \(|r| < 1\), it converges to \(\frac{a}{1-r}\).This is vital for power series as it provides a starting strategy for representing functions like \((1+x)^{-3}\).
  • The geometric series \(\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n\) is a classic example.
  • Its structure gives insight into more complex series through differentiation and algebraic manipulation.
By manipulating such series, we can express different functions as power series, allowing us to compute easily over a given interval.
Binomial Coefficients
Binomial coefficients are represented by \(\binom{n}{k}\), read as "n choose k." They arise in various power series expansions and binomial theorem applications.
They tell us how many ways we can choose \(k\) elements from a set of \(n\) elements, and they have mathematical significance in expanding binomials and more.
  • In the expression \((1+x)^{-k}\), coefficients \(\binom{k+n-1}{n}\) form the parts of the series.
  • For example, the binomial coefficient \(\binom{2+n}{n}\) was calculated as \(\frac{(2+n)!}{2!n!}\).
These coefficients are central to the power series representation as they reveal the contributing terms in the series, ensuring we can represent polynomial and rational functions accurately.

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Most popular questions from this chapter

Use the integral test (11.23) to determine the convergence or divergence of the series. $$ \sum_{n=1}^{x} \frac{1}{(3 n+2)^{3}} $$

Find a series representation for \(e^{-x}\) in powers of \(x+2\).

(a) Let \(g(x)\) be the sum of the first two nonzero terms of the Maclaurin series for \(f(x) .\) Use \(g(x)\) to approximate \(\int_{0}^{1} f(x) d x\) and \(\int_{1}^{2} f(x) d x .\) (b) Graph, on the same coordinate axes, \(f\) and \(g\) for \(0 \leq x \leq 2,\) and then use the graphs to compare the accuracy of the approximations in (a). \(f(x)=\sin \left(x^{2}\right)\)

Use the power series representation for \(\left(1-x^{2}\right)^{-1}\) to find a power series representation for \(2 x\left(1-x^{2}\right)^{-2}\).

Exer. \(21-22:\) For the given \(k,\) graph \(f(x)=(1+x)^{k}\) and \(g(x)=1+k x\) on the same coordinate plane for \(-1 \leq x \leq 1\). Use the graphs to estimate the largest closed interval on which \(|f(x)-g(x)| \leq \frac{1}{10}\). $$ k=\frac{1}{2} $$
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