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I need to calculate the following expression. Is there any explanation to convert this expression into normal expression without those letters for sum and the product? Just normal expression. $$... 1answer 66 views ### Is it possible to get a 'closed form' for \sum_{k=0}^{n} a_k b_{n-k}? This came up when trying to divide series, or rather, express \frac1{f(x)} as a series, knowing that f(x) has a zero of order one at x=0, and knowing the Taylor series for f(x) (that is ... 6answers 992 views ### Evaluating â€Ž\sum_{n=2}^{\infty}\frac{\zeta(n)}{k^n} â€ŽIf f\left(z \right)=\sum_{n=2}^{\infty}a_{n}z^n and \sum_{n=2}^{\infty}|a_n| converges thenâ€Ž,$$\sum_{n=1}^{\infty}f\left(\frac{1}{n}\right)=\sum_{n=2}^{\infty}a_n\zeta\left(n\right)â€Ž.$$â€ŽSince ... 1answer 138 views ### How to find a compact expression for \sum\limits_{k=1}^n \frac k{(K+1)!}?$$\sum_{k=1}^n \frac k{(K+1)!}$$How to find a compact expression? (Original scan here) 0answers 42 views ### Evaluation of a multiple sum involving \min\{i_0, i_1+ \cdots+ i_n\} with i_1+ \cdots+i_n\leq x How can I calculate \displaystyle\sum_{i_0=0}^x \sum_{i_1,\ldots, i_n=0}^1I_{i_1+ \cdots+i_n\leq x}\min\{i_0, i_1+ \cdots+ i_n\} as a function of n,x? I_{i_1+ \cdots+i_n\leq x} is ... 3answers 268 views ### Closed form of \sum\limits_{i=1}^n k^{1/i} or asymptotic equivalent when n\to\infty Is there a "closed form" for \displaystyle S_n=\sum_{i=1}^n k^{1/i} ? (I don't think so) If not, can we find a function that is asymptotically equivalent to S_n as n\to\infty ? 2answers 170 views ### Closed form of \sum\limits_{i=1}^n\left\lfloor\frac{n}{i}\right\rfloor^2? Does \displaystyle\sum_{i=1}^n\left\lfloor\dfrac{n}{i}\right\rfloor^2 admit a closed form expression? 4answers 564 views ### \sum_{n=1}^\infty(n\ \text{arccot}\ n-1) Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction.$$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$1answer 225 views ### Closed form for \sum_{n=1}^\infty\frac{(-1)^n n^a H_n}{2^n} Is there a closed form for the sum$$\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.$$This is a ... 3answers 460 views ### Closed form for \sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n} Please help me to find a closed form for the sum$$\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.$$3answers 200 views ### Calculate \sum\limits_{k=801}^{849}{ \binom {2400} {k}}  Is any formula which can help me to calculate directly the following sum :$$\sum_{k=801}^{849} \binom {2400} {k} \text{ ? } $$Or can you help me for an approximation? Thanks :) 2answers 121 views ### Calculation of binomial sum \displaystyle \sum_{r=1}^{n}r.\binom{n}{r}x^r.(1-x)^{n-r} = \;\;? [closed] How can I calculate$$\displaystyle \sum_{r=1}^{n} r \binom{n}{r}x^r (1-x)^{n-r} =\;\; ?$$0answers 195 views ### Double sum with binomial coefficients Find a closed form formula for this sum:$$\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}$$It's quite likely that it can be ... 4answers 97 views ### What is the formula of: a^{0} + a^{1} + a^{2} + … + a^{n-1} + a^{n}? [duplicate] What is the formula of:$$a^{0} + a^{1} + a^{2} + ... + a^{n-1} + a^{n}$$Any ideas? 2answers 76 views ### Evaluating a nested summation I am learning how to evaluate summation and got stuck on evaluating this summation: It would be great if you could help me through this.$$ \sum_{i=0}^{n} \sum_{i=0}^{m}3^{i+j} $$also after a bit ... 2answers 436 views ### Closed form sum of \sum^{\infty}_{n=1} \frac{1}{3^n-1} Wolframalpha uses q-Polygamma function to represent the sum, hence essentially does nothing. Here I wonder if this sum can be represented by elementary function. The summation is like a infinite ... 3answers 313 views ### Closed form for the sum of even fibonacci numbers? I recently took a look a project euler, and I am trying to think of a smart way to do number 2. I looked at the sequence, and I saw that the question is basically asking for$$ \sum_{i=1}^n F_{3i}  ...
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 $\sum^\infty_{i=1}ia^i$.