# Questions tagged [summation]

2,002 questions
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### Looking for a clever simplification of $\sum_{j=2}^{T_L}(j-1-O_L)(2j-n-1)+\sum_{i=T^U}^{n-1}(O^U-n+i)(2i-n-1)$

I have the expression (which I got by adding up all the distances between $n$ natural numbers bounded by $a_n$ and $a_1$): $$\sum_{j=2}^{T_L}(j-1-O_L)(2j-n-1)+\sum_{i=T^U}^{n-1}(O^U-n+i)(2i-n-1)$$ ...
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### Richert’s theorem breaks down for $n = 11$

In 1949 H.-E.Richert proved (1) that every positive integer typeset structure is a sum of distinct primes. For more information please look at (2), and (3). However, if you consider $n = 11$, you ...
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### Bizarre differential identity.

Let $d\ge 1$ be an integer. Let $m$ and $n$ be integers subject to $m \ge n+d-1$. The question is to prove the following identity. \sum\limits_{j=-1}^{d-1} \sum\limits_{\underset{d_1,...
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### Closed form expression for a sum

I want to calculate a sum of the form $$\sum_{k=0}^m \frac{\Gamma[m+1+\alpha-k]^2}{\Gamma[m+1-k]^2}\frac{\Gamma[x+k]}{\Gamma[x]k!}$$ where $m>0$ and belongs to integers and $\alpha$ takes half ...
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### Is this function monotonically non-decreasing?

I am wondering if the function $L[n]$ defined on $n=0,1,2,\ldots,N$ below is "monotonically" non-decreasing in $n$. I put monotonically in quotes because the function is not continuous and I am not ...
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### Product of Summations for All Subsets

We have a set $X$ of $n$ integers $\{$$x_1, x_2, .. , x_n$$\}$, for which there are $2^n$ total subsets. The summation $s$ of a subset $X'$ is simply the sum of all integers present in $X'$, ...
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### Showing that a number is part of sequence A000275 in OEIS

Consider the sequence of integers defined recursively by $c_0 = 1$ and $$c_p = \sum_{l = 0}^{p-1} (-1)^{p+l+1} \binom{p}{l}^2 c_l$$ for $p \geq 1$. This is sequence A000275 in the online ...
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### Why does the following equation hold?

$\sum_1^\infty\frac{(k\theta e^{-\theta})^k}{k!}=\frac{\theta}{1-\theta}$, where $0<\theta<1$. It can be verified via simulation, but I haven't proved it. Are there any previous results on ...
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### Show that $\sum x^p$ over primes must have a non-trivial zero.

The sum $\sum x^n$ is unbounded in $|x| \le 1$. Similarly if $p$ is prime then trivially $\sum x^p$ is also unbounded in $|x| < 1$ because all primes $> 2$ are odd so the lower bound would ...
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### Closed form and asymptotic solutionof $\sum_{a=1}^N \lfloor N/a \rfloor \lfloor (N \pm \lfloor N/a \rfloor)/a \rfloor$

I am solving polynomials with constraints on the coefficients and I get the following expression $$\sum_{a=1}^N \lfloor N/a \rfloor \lfloor (N \pm \lfloor N/a \rfloor)/a \rfloor$$. I am looking for ...
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### A summation of the multiplication of reflected Mobius functions and their behavior for different values of $k$

$$\Psi_k(N)=\sum_{n=1}^{N} \mu(n)\mu(k-n)$$ where $\mu(n)$ is the Mobius function. This function is interesting to me because for the case of $k=N$ it has the symmetric property of being odd with ...
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### Summation of Integrals without using $\zeta(2)$

How do you evaluate the sum $$\sum_{k=1}^\infty\int_{\sqrt k}^{\sqrt{k+1}} \left(\frac{x^2}{k}-1\right)\, dx$$ without using $\zeta(2)$? I can see some relationship to $\dfrac1{\lfloor x^2\rfloor}$ ...
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### A sort of Wolstenholme's $p^2$-congruence.

It is well-known (Wolstenholme Theorem) that for any prime $p$ such that $p>3$ the Harmonic number $H_{p-1}$ satisfies the congruence $$H_{p-1}:= \sum_{i=1}^{p-1}\frac{1}{i}\equiv 0 \pmod {p^2}$$ ...
I am told that $A$ and $B$ are random variables, and that $\mathbf{E}(A|B) = \gamma B$. Define $D = A/B$. Using the law of iterated expectations it can be shown that $\mathbf{E}(D) = \gamma$. Now ...