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Problem 3

Show that \(f_{n}(z)=\frac{z^{n}}{n}\) converges uniformly for \(|z|<1 .\) Show also that \(f_{n}^{\prime}(z)\) does not converge uniformly for \(|z|<1\) but it does converge uniformly for \(|z| \leq r\) for \(r<1\)

Expert verified

\(f_{n}(z)\) uniformly converges for \(|z|<1\), while \(f_{n}^{\prime}(z)\) does not for \(|z|<1\), but does uniformly converge for \(|z| \leq r\) when \(r<1\).

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Chapter 6

Assume that \(f\) is analytic for \(|z|

Chapter 6

Suppose \(\left\\{a_{n}\right\\}\) is a complex sequence whose partial sums \(\sum_{i=1}^{n} a_{i}\) are bounded. If \(\left\\{b_{n}\right\\}\) is a real sequence that is monotonically decreasing to 0 , show that \(\sum_{n=1}^{\infty} a_{n} b_{n}\) converges.

Chapter 6

Suppose \(\sum_{n=0}^{\infty} a_{n} z^{n}\) has radius of convergence $R,
0

Chapter 6

Show that \(f_{n}=u_{n}+i v_{n}\) converges uniformly to \(f=u+i v\) if and only if \(\left\\{u_{n}\right\\}\) converges uniformly to \(u\) and \(\left\\{v_{n}\right\\}\) converges uniformly to \(v\).

Chapter 6

Let \(s_{n}=\sum_{k=1}^{n} a_{k} .\) If \(a_{k}=1 / k\), show that $s_{2 n+1}-s_{2^{n}}>\frac{1}{2}\( for every \)n .$ If \(a_{k}=(-1)^{k+1} / k\), show that \(\left|s_{n+p}-s_{n}\right|<2 / n\) for every \(n\) and \(p .\)

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