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Chapter 8: Problem 33
Find the sum of the given series. $$ \sum_{n=0}^{\infty} 9(0.1)^{n} $$
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It is known that \(\sum_{n=1}^{\infty} \frac{1}{n^{2}}=\frac{1}{6} \pi^{2}\) and \(\sum_{n=1}^{\infty} \frac{1}{n^{4}}=\frac{1}{90} \pi^{4}\). Let \(f(x, y)=\sum_{n=1}^{\infty} \frac{x+y \cdot(\pi n)^{2}}{n^{4}} .\) Describe the solution set \(\left\\{(x, y) \in \mathbb{R}^{2}: f(x, y)=0\right\\}\).
It is known that \(\sum_{n=1}^{\infty} 1 / n^{2}=\pi^{2} / 6 .\) What are the values of the convergent series $$ 1 / 2^{2}+1 / 4^{2}+1 / 6^{2}+1 / 8^{2}+\cdots $$ and $$ 1+1 / 3^{2}+1 / 5^{2}+1 / 7^{2}+\cdots ? $$
In each of Exercises 66-71, evaluate the given series (exactly). $$ 1-\frac{1}{1 !}+\frac{1}{2 !}-\frac{1}{3 !}+\frac{1}{4 !}-\cdots $$
If \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) and \(g(x)=\sum_{n=0}^{\infty} b_{n} x^{n}\) converge on \((-R, R),\) then we may formally multiply the series as though they were polynomials. That is, if \(h(x)=f(x) g(x),\) then $$ h(x)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n} a_{k} b_{n-k}\right) x^{n} $$ The product series, which is called the Cauchy product, also converges on \((-R, R) .\) Exercises \(59-62\) concern the Cauchy product. Suppose that the series \(\sum_{n=0}^{\infty} a_{n} x^{n}\) converges on \((-R, R)\) to a function \(f(x)\) and that \(|f(x)| \geq k>0\) on that interval for some positive constant \(k\). Then, \(1 / f(x)\) also has a convergent power series expansion on \((-R, R) .\) Compute its coefficients in terms of the \(a_{n}\) 's. Hint: Set $$ \frac{1}{f(x)}=g(x)=\sum_{n=0}^{\infty} b_{n} x^{n} $$ Use the equation \(f(x) \cdot g(x)=1\) to solve for the \(b_{n}\) 's.
Let \(\left\\{a_{n}\right\\}\) be a sequence of positive numbers. In a course on mathematical analysis, one learns that if the two limits \(\lim _{n \rightarrow \infty} a_{n+1} / a_{n}\) and \(\lim _{n \rightarrow \infty} a_{n}^{1 / n}\) exist, then they are equal. In each of Exercises \(65-68\), produce a plot that illustrates the equality of these two limits. Your plot should include a horizontal line that is the asymptote of the points \(\left\\{\left(n, a_{n+1} / a_{n}\right)\right\\}\) and \(\left\\{\left(n, a_{n}^{1 / n}\right)\right\\}\). \(a_{n}=e^{n} / \ln (1+n)\)
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