0
votes
0answers
40 views

Is there a closed form of the following expression.

Does anyone know of a way to write a closed form of the following expression using only addition, subtraction, multiplication, and division? $$ \left[\sum\limits_{i=1}^n \ln(i)\right]^e $$
0
votes
1answer
26 views

Close form of a power series starting at $n=2$

This is the power series I am looking at $\sum_{n=2}^{\infty}{n(n-1)z^n}$. I want to find the closed form of this power series. This is my approach, if I divide the power series by $z^2$, then I ...
0
votes
1answer
56 views

How to solve this summation (Lerch Transcendent)?

How is it possible to deduce the closed form of the following? $$\sum_{i = 0}^{n - 1} \frac{2^i}{n - i} = ?$$
2
votes
0answers
55 views

Find the sum of exponentails of squares $\sum_{r=1}^n e^{-\alpha r^2}$

I would like to find $$a_n =\sum_{r=1}^n e^{-\alpha r^2},\qquad \alpha\in\mathbb{R}$$ I tried to solve the equivalent recursion $$a_n=a_{n-1}+e^{-\alpha n^2}\quad(n>0),\qquad a_0=0.$$ with an ...
3
votes
2answers
108 views

Finding the closed form of a sum

I would like to find the closed form of the sum $\sum_{n = 4}^{x}(x - n)$. I believe that the derivative is $x - 4$, but when I take the integral of that and graph it, the sum and $\frac{x^2}{2} + 4x$ ...
2
votes
3answers
64 views

Closed form for a trigonometric partial sum

I know that: $$\sum_{k=1}^n\arctan(2k^2)=\frac{\pi n}{2}-\frac{1}{2}\arctan(\frac{2n(n+1)}{2n+1})$$ Can a similar closed form expressions be given for $\sum_{k=1}^n \arctan(k^2)$? I was able to ...
2
votes
1answer
80 views

Evaluate $\sum_{n=1}^{\infty} \frac{\ln n}{(n+1)!}$

Is there a closed form expression for this sum (or very similar sums) with known transcendental functions or irrational numbers? $$\sum_{n=1}^{\infty} \frac{\ln n}{(n+1)!} \approx ...
0
votes
1answer
39 views

The closed form of a sum of mod(k,m) where k goes from 1 to a arbitrary number.

Is there a closed form for $\sum_{n=0}^{C} mod(n,m)$ for arbitrary integers m ?
0
votes
2answers
87 views

How can I compute this sum of binomial

Is there any way to compute the following sum: $\displaystyle{ \sum_{\ell = {n + 1 \over{\vphantom{\LARGE A}2}}}^{n}{n \choose \ell}5^{n - \ell}}$ where $n$ is odd. Thank you.
0
votes
0answers
54 views

Close formula for triple sum binomial coefficient

I need to compute the following sum or to find a lower and upper bound that limit the sum: $\sum_{l=\frac{n+1}{2}}^n \binom{n}{l} \sum_{t=0}^{n-l} \binom{l}{t} 2^{l-t} \sum_{m=t}^{n-l} \binom{n-l}{m} ...
1
vote
1answer
64 views

Is it possible to get a 'closed form' for $\sum_{k=0}^{n} a_k b_{n-k}$?

This came up when trying to divide series, or rather, express $\frac1{f(x)}$ as a series, knowing that $f(x)$ has a zero of order one at $x=0$, and knowing the Taylor series for $f(x)$ (that is ...
21
votes
6answers
794 views

Evaluating $‎\sum_{n=2}^{\infty}\frac{\zeta(n)}{k^n}$

‎If $f\left(z \right)=\sum_{n=2}^{\infty}a_{n}z^n$ and $\sum_{n=2}^{\infty}|a_n|$ converges then‎, $$\sum_{n=1}^{\infty}f\left(\frac{1}{n}\right)=\sum_{n=2}^{\infty}a_n\zeta\left(n\right)‎.$$ ‎Since ...
0
votes
1answer
122 views

How to find a compact expression for $\sum\limits_{k=1}^n \frac k{(K+1)!}$?

$$\sum_{k=1}^n \frac k{(K+1)!}$$ How to find a compact expression? (Original scan here)
9
votes
3answers
248 views

Closed form of $\sum\limits_{i=1}^n k^{1/i}$ or asymptotic equivalent when $n\to\infty$

Is there a "closed form" for $\displaystyle S_n=\sum_{i=1}^n k^{1/i}$ ? (I don't think so) If not, can we find a function that is asymptotically equivalent to $S_n$ as $n\to\infty$ ?
2
votes
2answers
150 views

Closed form of $\sum\limits_{i=1}^n\left\lfloor\frac{n}{i}\right\rfloor^2$?

Does $\displaystyle\sum_{i=1}^n\left\lfloor\dfrac{n}{i}\right\rfloor^2$ admit a closed form expression?
29
votes
4answers
532 views

$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$

Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction. $$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
16
votes
1answer
195 views

Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^a H_n}{2^n}$

Is there a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^a H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$ This is a ...
19
votes
3answers
412 views

Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}$

Please help me to find a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$
5
votes
3answers
191 views

Calculate $\sum\limits_{k=801}^{849}{ \binom {2400} {k}} $

Is any formula which can help me to calculate directly the following sum : $$\sum_{k=801}^{849} \binom {2400} {k} \text{ ? } $$ Or can you help me for an approximation? Thanks :)
-2
votes
2answers
108 views

Calculation of binomial sum $\displaystyle \sum_{r=1}^{n}r.\binom{n}{r}x^r.(1-x)^{n-r} = \;\;?$ [closed]

How can I calculate $$\displaystyle \sum_{r=1}^{n} r \binom{n}{r}x^r (1-x)^{n-r} =\;\; ?$$
1
vote
0answers
150 views

Double sum with binomial coefficients

Find a closed form formula for this sum: $$\sum_{1\le i<j\le m} \sum_{\substack{1\le k,l\le n \\ k+l\le n}}{n\choose k} {n-k\choose l} (j-i-1)^{n-k-l}$$ It's quite likely that it can be ...
1
vote
4answers
95 views

What is the formula of: $a^{0} + a^{1} + a^{2} + … + a^{n-1} + a^{n}$? [duplicate]

What is the formula of: $$a^{0} + a^{1} + a^{2} + ... + a^{n-1} + a^{n}$$ Any ideas?
0
votes
2answers
68 views

Evaluating a nested summation

I am learning how to evaluate summation and got stuck on evaluating this summation: It would be great if you could help me through this. $$ \sum_{i=0}^{n} \sum_{i=0}^{m}3^{i+j} $$ also after a bit ...
4
votes
2answers
385 views

Closed form sum of $\sum^{\infty}_{n=1} \frac{1}{3^n-1}$

Wolframalpha uses $q$-Polygamma function to represent the sum, hence essentially does nothing. Here I wonder if this sum can be represented by elementary function. The summation is like a infinite ...
5
votes
3answers
247 views

Closed form for the sum of even fibonacci numbers?

I recently took a look a project euler, and I am trying to think of a smart way to do number 2. I looked at the sequence, and I saw that the question is basically asking for $$ \sum_{i=1}^n F_{3i} $$ ...
3
votes
1answer
678 views

Trouble finding closed-form solution to summation

I have this question on my assignment for a computer science course (analysis of algorithms), so any help would be appreciated, but I am not looking for the answer itself. I am trying to find the ...
0
votes
5answers
4k views

Closed form expression of infinite summation

What is the idea behind a closed form expression and what is the general way of finding the closed form solution of an infinite summation? context: closed form solution of $\sum^\infty_{i=1}ia^i$.