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Chapter 7: Problem 61

State the guidelines for finding a Taylor series.

Short Answer

Expert verified
A Taylor series represents a function as an infinite sum of terms that are calculated from the function's derivatives at a certain point. The coefficient of each term is the nth derivative of the function evaluated at the point, divided by \(n!\). The Taylor series for a function at \(a\) is \(f(a) + f'(a) \cdot (x-a)/1! + f''(a) \cdot (x-a)^2/2! + f'''(a) \cdot (x-a)^3/3! + \cdots\). Finally, the radius of convergence represents the largest disk in which the series converges.

Step by step solution

01

Understand the Concept

The Taylor series of a function is an infinite series of terms that are expressed as the function's derivatives at a certain point. It's a way of representing a function as an infinite sum that is used to approximate functions in mathematics and engineering.
02

Formula for Taylor Series

The Taylor series of a function \(f(x)\) that is infinitely differentiable at a real or complex number \(a\) is given by \[f(a) + f'(a) \cdot (x-a)/1! + f''(a) \cdot (x-a)^2/2! + f'''(a) \cdot (x-a)^3/3! + \cdots\]
03

Calculate the Coefficients

The coefficient of each term in the Taylor series is the function's nth derivative evaluated at the number a, divided by \(n!\). This is also known as the Taylor coefficient.
04

Understand the Radius of Convergence

The radius of convergence of a power series in the complex plane is either a nonnegative real number or 鈭� which represents the radii of the largest disk in which the series converges. The series might also converge on the boundary of the disk.

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Most popular questions from this chapter

Find the sum of the convergent series. $$ 3-1+\frac{1}{3}-\frac{1}{9}+\cdots $$

Use a graphing utility to graph the function. Identify the horizontal asymptote of the graph and determine its relationship to the sum of the series. $$ \frac{\text { Function }}{f(x)=2\left[\frac{1-(0.8)^{x}}{1-0.8}\right]} \frac{\text { Series }}{\sum_{n=0}^{\infty} 2\left(\frac{4}{5}\right)^{n}} $$

(a) You delete a finite number of terms from a divergent series. Will the new series still diverge? Explain your reasoning. (b) You add a finite number of terms to a convergent series. Will the new series still converge? Explain your reasoning.

Find the sum of the convergent series. $$ \sum_{n=1}^{\infty} \frac{1}{9 n^{2}+3 n-2} $$

Let \(\left\\{a_{n}\right\\}\) be a monotonic sequence such that \(a_{n} \leq 1\). Discuss the convergence of \(\left\\{a_{n}\right\\} .\) If \(\left\\{a_{n}\right\\}\) converges, what can you conclude about its limit?
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