# Questions tagged [summation]

11,135 questions
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### When is $a(n)$ prime?

Question: When is $a(n)\in P$ compared to all possible values of $n$? where $P$ denotes the set of primes. What is the density of the primes in the sequence? Consider the sum of the prime counting ...
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### Approximating an infinite sum: $\sum_t t^d \exp(-(t-1)^2/2)$

I am seeking to upper bound the limit of the following infinite summation, when a free parameter $\beta$ can be chosen, perhaps dependent on $d$, to help reduce the sum: f(\beta,d) = \...
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### Derive the sum of $\sum_{i=1}^n ix^{i-1}$

For the series $$1 + 2x + 3x^2 + 4x^3 + 5x^4 + ... + nx^{n-1}+...$$ and $x \ne 1, |x| < 1$. I need to find partial sums and finally, the sum $S_n$ of series. Here is what I've tried: We ...
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### Proving $\sum_{j=0}^{N-1}\cos\frac{\left(2j+1\right)\pi}{2N}=0$ [duplicate]

Let $l\in\mathbb{Z}$ and $N\in\mathbb{N}$. I need to prove the following: $$\sum_{j=0}^{N-1}\cos\left(l\frac{\left(2j+1\right)\pi}{2N} \right)=0$$ I tried to use Euler ...
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### Find the Sum of the Series: $\sum_{0}^{\infty}\frac{(-1)^n\pi^{2n}}{6^{2n}(2n)!}$

Find the Sum of the Series $$\sum_{n=0}^{\infty}\frac{(-1)^n\pi^{2n}}{6^{2n}(2n)!}$$ Alright, so I think I may have gotten this problem correct but I'm a little hesitant, so If you could check my ...
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### Find the partial sum formula of $\sum_{i=1}^n \frac{x^{2^{i-1}}}{1-x^{2^i}}$

Given next series: $$\frac{x}{1 - x^2} + \frac{x^2}{1 - x^4} + \frac{x^4}{1 - x^8} + \frac{x^8}{1 - x^{16}} + \frac{x^{16}}{1 - x^{32}} + ...$$ and $|x| < 1$. Need to derive $S_n$ formula ...
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### What is $\sum_{k=1}^m\left[2^k\begin{pmatrix} n \\ k \end{pmatrix}\right]^2$?

The equations $$\sum_{k=1}^m\begin{pmatrix} n \\ k \end{pmatrix}^2 \quad \text{and} \quad \sum_{k=1}^m\left[2^k\begin{pmatrix} n \\ k \end{pmatrix}\right]^2$$ popped up in some of my calculations, ...
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### Is it incorrect to have a sum of an infinite weighted set?

I am currently in a revision of a paper. I have found something that I would only like to change if it is currently really notationally false because each change bears a few risks. What I currently ...
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### Sum of sequential numbers rational or irrational? [closed]

Is following infinite sum is rational or irrational? $\sum_{n=1}^\infty n/10^n$