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Chapter 9: Problem 79
Find a power series that has (2,6) as an interval of convergence.
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Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than \(10^{-4}.\) $$\int_{0}^{0.2} \frac{\ln (1+t)}{t} d t$$
If the power series \(f(x)=\sum c_{k} x^{k}\) has an interval of convergence of
\(|x|
Recall that the Taylor series for \(f(x)=1 /(1-x)\) about 0 is the geometric series \(\sum_{k=0}^{\infty} x^{k} .\) Show that this series can also be found as a binomial series.
There are several proofs of Taylor's Theorem, which lead to various forms of the remainder. The following proof is instructive because it leads to two different forms of the remainder and it relies on the Fundamental Theorem of Calculus, integration by parts, and the Mean Value Theorem for Integrals. Assume that \(f\) has at least \(n+1\) continuous derivatives on an interval containing \(a\) a. Show that the Fundamental Theorem of Calculus can be written in the form $$f(x)=f(a)+\int_{a}^{x} f^{\prime}(t) d t$$ b. Use integration by parts \(\left(u=f^{\prime}(t), d v=d t\right)\) to show that $$f(x)=f(a)+(x-a) f^{\prime}(a)+\int_{a}^{x}(x-t) f^{\prime \prime}(t) d t$$ c. Show that \(n\) integrations by parts gives $$ \begin{aligned} f(x)=& f(a)+\frac{f^{\prime}(a)}{1 !}(x-a)+\frac{f^{\prime \prime}(a)}{2 !}(x-a)^{2}+\cdots \\ &+\frac{f^{(n)}(a)}{n !}(x-a)^{n}+\underbrace{\int_{a}^{x} \frac{f^{(n+1)}(t)}{n !}(x-t)^{n} d t}_{R_{n}(x)} \end{aligned} $$ d. Challenge: The result in part (c) looks like \(f(x)=p_{n}(x)+\) \(R_{n}(x),\) where \(p_{n}\) is the \(n\) th-order Taylor polynomial and \(R_{n}\) is a new form of the remainder, known as the integral form of the remainder. Use the Mean Value Theorem for Integrals (Section 5.4 ) to show that \(R_{n}\) can be expressed in the form $$ R_{n}(x)=\frac{f^{(n+1)}(c)}{(n+1) !}(x-a)^{n+1} $$ where \(c\) is between \(a\) and \(x\)
Symmetry a. Use infinite series to show that \(\cos x\) is an even function. That is, show \(\cos (-x)=\cos x.\) b. Use infinite series to show that \(\sin x\) is an odd function. That is, show \(\sin (-x)=-\sin x.\)
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