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Chapter 11: Problem 11
Use deMoivre's Theorem to write \(\sin ^{3} \theta\) in terms of \(\sin \theta\) and \(\sin 3 \theta .\)
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Do the following: a. Compute: \(\lim _{n \rightarrow \infty} n \ln \left(1-\frac{3}{n}\right)\). b. Use L'Hopital's Rule to evaluate \(L=\lim _{x \rightarrow \infty}\left(1-\frac{4}{x}\right)^{x}\). [Hint: Consider \(\ln L\).] c. Determine the convergence of \(\sum_{n=1}^{\infty}\left(\frac{n}{3 n+2}\right)^{n^{2}}\). d. Sum the series \(\sum_{n=1}^{\infty}\left[\tan ^{-1} n-\tan ^{-1}(n+1)\right]\) by first writing the \(N\) th partial sum and then computing \(\lim _{N \rightarrow \infty} s_{N}\).
Recall that the alternating harmonic series converges conditionally. a. From the Taylor series expansion for \(f(x)=\ln (1+x)\), inserting \(x=1\) gives the alternating harmonic series. What is the sum of the alternating harmonic series? b Because the alternating harmonic series does not converge absolutely, a rearrangement of the terms in the series will result in series whose sums vary. One such rearrangement in alternating \(p\) positive terms and \(n\) negative terms leads to the following sum \(^{10}\) : $$ \begin{gathered} \frac{1}{2} \ln \frac{4 p}{n}=\underbrace{\left(1+\frac{1}{3}+\cdots+\frac{1}{2 p-1}\right)}_{p \text { terms }}-\underbrace{\left(\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2 n}\right)}_{n \text { terms }} \\ +\underbrace{\left(\frac{1}{2 p+1}+\cdots+\frac{1}{4 p-1}\right)}_{p \text { terms }}-\underbrace{\left(\frac{1}{2 n+2}+\cdots+\frac{1}{4 n}\right)}_{n \text { terms }}+\cdots . \end{gathered} $$ Find rearrangements of the alternating harmonic series to give the following sums; that is, determine \(p\) and \(n\) for the given expression and write down the above series explicitly; that is, determine \(p\) and \(n\) leading to the following sums. i. \(\frac{5}{2} \ln 2\). ii. \(\ln 8\). iii. \(0 .\) iv. A sum that is close to \(\pi\).
Find the sum for each of the series: a. \(\sum_{n=0}^{\infty} \frac{(-1)^{n} 3}{4^{n}}\) b. \(\sum_{n=2}^{\infty} \frac{2}{5^{n}}\). c. \(\sum_{n=0}^{\infty}\left(\frac{5}{2^{n}}+\frac{1}{3^{n}}\right)\). d. \(\sum_{n=1}^{\infty} \frac{3}{n(n+3)} .\)
Determine the radius and interval of convergence of the following infinite series: a. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{(x-1)^{n}}{n}\) b. \(\sum_{n=1}^{\infty} \frac{x^{n}}{2^{n} n !}\) c. \(\sum_{n=1}^{\infty} \frac{1}{n}\left(\frac{x}{5}\right)^{n}\) d. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{x^{n}}{\sqrt{n}}\)
Consider Gregory's expansion $$ \tan ^{-1} x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\cdots=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{2 k+1} x^{2 k+1} $$ a. Derive Gregory's expansion using the definition $$ \tan ^{-1} x=\int_{0}^{x} \frac{d t}{1+t^{2}} $$ expanding the integrand in a Maclaurin series, and integrating the resulting series term by term. b. From this result, derive Gregory's series for \(\pi\) by inserting an appropriate value for \(x\) in the series expansion for \(\tan ^{-1} x\).
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