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Problem 4

Let \(a_{n}>0\), and suppose \(\sum_{n=1}^{\infty} a_{n}\) converges. If \(r_{n}=\sum_{k=n}^{\infty} a_{k}\), show that (a) \(\sum_{n=1}^{\infty} \frac{a_{n}}{r_{n}}\) diverges (b) \(\sum_{n=1}^{\infty} \frac{a_{n}}{\sqrt{r_{n}}}\) converges.

Expert verified

(a) The series \(\sum_{n=1}^{\infty} \frac{a_{n}}{r_{n}}\) diverges according to the limit comparison test. (b) The series \(\sum_{n=1}^{\infty} \frac{a_{n}}{\sqrt{r_{n}}}\) converges also according to the limit comparison test.

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Chapter 6

Suppose \(\left\\{a_{n}\right\\}\) is a sequence of integers. Prove that \(\sum_{n=0}^{\infty} a_{n} z^{n}\) is either an entire function or has radius of convergence at most one.

Chapter 6

Write Taylor expansions for the polynomial \(P(z)=z^{3}+3 z^{2}-2 z+1\) in powers of (a) \(z-1\) (b) \(z+2\) (c) \(z-i\).

Chapter 6

Show that \(\lim \sup _{n \rightarrow \infty} a_{n}=L, L\) finite, if and only
if the following conditions hold: For any \(\epsilon>0\),
a) \(a_{n}

Chapter 6

Show that \(\lim \inf _{n \rightarrow \infty} a_{n}=L, L\) finite, if and only
if the following conditions hold: For any \(\epsilon>0\),
a) \(a_{n}

Chapter 6

Suppose that \(a_{n}+A a_{n-1}+B a_{n-2}=0(n=2,3,4, \ldots) .\) Show that $$ \sum_{n=0}^{\infty} a_{n} z^{n}=\frac{a_{0}+\left(a_{1}+a_{0} A\right) z}{1+A z+B z^{2}} $$ at all points where the power series converges. What is the radius of convergence?

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