# Tagged Questions

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If the root test limit tends to infinity, is that sufficient to say we have an infinite radius of convergence? Thanks
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### Power series difficulty

How would I find the region of convergence of the series of $\frac{1}{n^3}(\frac{z+1}{z-1})^n$. I thought about rewriting $\frac{z+1}{z-1}$ as $\frac{2}{z-1}+1$ but I don't think that helps. Thanks
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Find the radius of convergence of the series of $\frac{2^n(4z-8)^n}{n}$ My answer: $(4z-8)^n=4^n(z-2)^n=2^{2n}(z-2)^n$. Let $c_{n}=\frac{2^{3n}}{n}$. Then $\frac{c_{n}}{c_{n+1}}=\frac{n+1}{2n}$ so ...
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### Divergent Complex sequence with distinct points and without accumulation point tends to $\infty$?

Is it true that any Divergent complex sequence with distinct points and without accumulation point tends to $\infty$ ? (One can also replace distinctness condition by condition finitely repeating ...
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### Series that diverges at infinitely many points on the unit circle

Initial problem Given $A=\{\alpha_1,\dots,\alpha_k\}$ with $|\alpha_i|=1$, does there exist a power series $\sum a_nz^n$ that converges everywhere on the unit circle except when $z\in A?$ ...
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### Change of Double Sum to Single Sum for Large N

I am reading article on X-ray interference. In the article authors claim that it is possible to replace double sum $\frac{1}{N} \sum_{n=1}^{N} \sum_{k=1}^{N} e^{-i \phi_{n}} e^{i \phi_{k}}$ by ...
### $\sin(1+\frac{1}{z-1})$ expanded in powers of $z-1$
The whole problem: Obtain the following Laurent expansions. State the first four nonzero terms. State explicitly the $n$th term in the series, and state the largest possible annular domain in which ...