# Tagged Questions

For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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### Complex numbers and geometric series

a) Use the formula for the sum of a geometric series to show that $$\sum _{k=1}^n\:\left(z+z^2+\cdots+z^k\right)=\frac{nz}{1-z}-\frac{z^2}{\left(1-z\right)^2}\left(1-z^n\right),\:z\ne 1$$ I thought ...
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### Abel's test and Leibnitz's test

Hi does anyone know how Abel's test and Leibnitz's test( also called the 'alternating series test' for convergence) are related? Is the alternating series test sometimes called Abel's test? The ...
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### Logarithmic Series Evaluation

I was trying to generate a direct formula for this series but I am not sure whether it is possible to do so. $$1\ln(1) + 2\ln(2) + 3\ln(3) + 4\ln(4)+\dots+(n-1)\ln(n-1) + n\ln(n)$$
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### Convergence of the sequence $\alpha_n = \frac{1}{\sqrt{n^2 + 1}} + \dots+\frac{1}{\sqrt{n^2+n}}$

How to determine the convergence of this sequence? $$\alpha_n = \frac{1}{\sqrt{n^2 + 1}} + \frac{1}{\sqrt{n^2+2}}+ \dots +\frac{1}{\sqrt{n^2+n}}$$ I was trying to show first that the sequence has a ...
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### Computing $\lim_{n\to\infty} \frac{\sqrt{n}}{4^{n}}\sum_{k=1}^{n} \binom{2n-1}{n-k}\frac{ 1}{(2k-1)^2+\pi^2}$

What tools would you recommend me for computing the limit below? $$\lim_{n\to\infty} \frac{\sqrt{n}}{4^{n}}\sum_{k=1}^{n}\frac{\displaystyle \binom{2n-1}{n-k}}{(2k-1)^2+\pi^2}$$ As soon as any ...
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### Find a series $f(r)=\sum_{0}^{\infty}a_nr^n$ s.t converges for $|r| < R$ and s.t. $\lim_{r\rightarrow R-} f(r)$ exists but series does not converge.

Find a series $f(r)=\sum_{0}^{\infty}a_nr^n$ s.t converges for $|r| < R$ and s.t. $\lim_{r\rightarrow R-} f(r)$ exists but series does not converge. How do I approach this? I think that $a_n r^n$ ...
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### Clarification: Proof of the quotient rule for sequences

My Problem I am currently looking for a proof for the quotient rule for sequences: $a_n$ and $b_n$ are two sequences with the limes a,b. So: When $a_n \rightarrow a$ and $b_n \rightarrow b$ ...
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### equivalent metrics and uniform equivalent metrics

Let (X,d) be the Euclidean metric on the real number, and define σ(x,y)=min{1,d(x,y)} if if x, y are rationals or x, y are irrationals, and σ(x,y)=1 otherwise. I would like to study if these metric ...
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### Proving another digammabinomial series result

This series is related to some extent to the previous question of mine, that is Computing $\sum_{n=1}^{\infty} \frac{\psi\left(\frac{n+1}{2}\right)}{ \binom{2n}{n}}$, where an approach by series only ...