# Tagged Questions

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### Help on a tough summation from Rudin?

I'm having a tough time deriving (4) from the bracketed expression in (3) shown in the photo. I've been futzing with partial sums of geometric series and binomial expansions for a while now with no ...
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### Application of Residue Theorem and limits

I am trying the following problem from Fisher's Complex Variables book: If $f$ is analytic on a plane except at poles $\gamma_1, \cdots \gamma_N$ and none of them are integers and ...
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### How to evaluate the summation $S_b$

This question is from my notebook, not hw or else, only exercise to understand better. I tried by myself. However, since my trail are too trivial, I dont need to write here. i am confused a bit. I ...
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### Sum $\sum_{n=0}^{N} z^n=\frac{1-z^{N+1}}{1-z}$

My professor just gave this as given in class today and went onto something else. Show that $$\sum_{n=0}^{N} z^n=\frac{1-z^{N+1}}{1-z}.$$ I would like to know how he obtained this neat expression ...
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### Simplifying a sum?

Define polynomials $P_{j,s}^{(r)}$ via the generating series $$\left(\frac{d^s}{dz^s}f(z)\right)^r=\sum_{j=0}^{\infty} P_{j,s}^{(r)}z^j,$$ where $r\geq 1$. Here, $f(z)=z+a_2z^2+a_3z^3+\cdots.$ I was ...
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### An identity about the Pi and Riemann's zeta function

How to prove the following identity? $$\sum_{n=1}^{\infty}\frac{(m-1)^n-1}{m^n}\zeta(n+1)=\pi\cot\left(\frac{\pi}{m}\right)$$
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### Proving $\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$

Wikipedia informs me that $$S = \vartheta(0;i)=\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$$ I tried considering $f(x,n) = e^{-x n^2}$ so that its ...
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### Prove that for n~=n' sum is much smaller than the case with n=n'

Hi I want to prove that this summation is much smaller for $n\neq n'$ than for the case where $n=n'$. I have seen this fact with simulation results. But I don't know how to prove it in mathematics. ...
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### Value of Summation of $\log(n)$

Context: I am learning Dijstra's Algorithm to find shortest path to any node, given the start node. Here, we can use Fibonnacci Heap as Priority Queue. Following is few lines of algorithm: ...
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### How to calculate $\sum_{k=1}^{k=n}\frac{\sin(kx)}{\sin^{k}(x)}$?

I was given an exercise: Calculate 1+$\sum_{k=1}^{k=n}\frac{\sin(kx)}{\sin^{k}(x)}$ I recognize $$\sin(kx)=Im(cis(kx))=Im(cis^{k}(x))$$ and $$\sin^{k}(x)=(Im(cis(x)))^{k}$$ but I do not know ...
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### Proving $\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$

I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$ I have some progress made, but I am stuck and could use some help. What I did: ...
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### Complex analysis equality in a limited sum

Let $z=e^{i\theta}$ with $\theta \in [0,2\pi[$ . Consider the sum $$\sum_{n=1}^{N} (e^{i\theta})^n.$$ How could this be equal to $$\frac{1-e^{iN+T\theta}}{1-e^{i\theta}} \quad ?$$ I tried to ...
I would to prove that $$1+\cos \theta+\cos 2\theta+\ldots+\cos n\theta =\displaystyle\frac{1}{2}+ \frac{\sin\left[(2n+1)\frac{\theta}{2}\right]}{2\sin\left(\frac{\theta}{2}\right)}$$ given that ...
### Proving the identity $\sum_{n=-\infty}^\infty e^{-\pi n^2x}=x^{-1/2}\sum_{n=-\infty}^\infty e^{-\pi n^2/x}.$
Can you help prove the functional equation: $$\sum_{n=-\infty}^\infty e^{-\pi n^2x}=x^{-1/2}\sum_{n=-\infty}^\infty e^{-\pi n^2/x}.$$ Specifically, I am looking for a solution using complex ...