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Chapter 9: Problem 6
If \(a\) and \(b\) are positive numbers, then \(\sum(a n+b)^{-p}\) converges if \(p>1\) and diverges if \(p \leq 1\).
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Give an example of a function which is equal to its Taylor series expansion about \(x=0\) for \(x \geq 0\), but which is not equal to this expansion for \(x<0\)
If \(\alpha \in \mathbb{R}\) and \(|k|<1\), the integral \(F(\alpha, k):=\int_{0}^{\alpha}\left(1-k^{2}(\sin x)^{2}\right)^{-1 / 2} d x\) is called an elliptic integral of the first kind. Show that $$F\left(\frac{\pi}{2}, k\right)=\frac{\pi}{2} \sum_{n=0}^{\infty}\left(\frac{1 \cdot 3 \cdots(2 n-1)}{2 \cdot 4 \cdots 2 n}\right)^{2} k^{2 n} \quad \text { for } \quad|k|<1$$
Show that if the partial sums \(s_{n}\) of the series \(\sum_{k=1}^{\infty} a_{k}\) satisfy \(\left|s_{n}\right| \leq M n^{r}\) for some \(r<1\), then the series \(\sum_{n=1}^{N} a_{n} / n\) converges.
If the partial sums of \(\sum a_{n}\) are bounded, show that the series \(\sum_{n=1}^{\infty} a_{n} e^{-n t}\) converges for \(t>0\).
If \(\left(a_{n}\right)\) is a decreasing sequence of strictly positive aumbers and if \(\sum a_{n}\) is convergent, show that \(\lim \left(n a_{n}\right)=0\)
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