Chapter 10: Problem 7
What is the interval of convergence of the power series \(\Sigma(4 x)^{k} ?\)
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Replace \(x\) by \(x-1\) in the series \(\ln (1+x)=\sum_{k=1}^{\infty} \frac{(-1)^{k+1} x^{k}}{k}\) to obtain a power series for \(\ln x\) centered at \(x=1 .\) What is the interval of convergence for the new power series?
Identify the functions represented by the following power series. $$\sum_{k=0}^{\infty} 2^{k} x^{2 k+1}$$
a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. $$f(x)=\frac{e^{x}+e^{-x}}{2}$$
Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Use the Taylor series. $$(1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots, \text { for }-1 < x < 1$$ $$(1+4 x)^{-2}$$
Find the remainder in the Taylor series centered at the point a for the following functions. Then show that \(\lim _{n \rightarrow \infty} R_{n}(x)=0\) for all \(x\) in the interval of convergence. $$f(x)=e^{-x}, a=0$$
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