# Tagged Questions

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### Simplifying big expression

What to do with this? $$f(x) = \frac{\sinh(\pi)}{\pi} + \frac{2\sinh(\pi)}{\pi}\sum_{n=1}^\infty (-1)^n \left[\frac{\cos(nx)-n \sin(nx)}{1 + n^2}\right]$$ Can it be simplified?
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### Closed form for this incomplete gamma series?

The series I'm working with is $$\sum_{k=0}^\infty \binom{z}{k}(-1)^k ( 1-\frac{\Gamma(k,-\log n)}{\Gamma(k)} )$$ with $z$ a complex variable and $\Gamma(k, -\log n)$ the upper incomplete gamma ...
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### Given a positive integer $k$, find the integer part of $n^2 /k$ for $n\ge 1$, and a related question.

For a given positive integer $k,$ I am looking for possible answers / literature about the sequence $(a_n)=([\frac{n^2}{k}])_{n=1}^\infty$, where $[x]=$the integer part of $x.$ This question is ...
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### Evaluation of a class of continued fractions

Is there a closed-form way of writing the continued fraction: $$1 + \frac{2}{3+ \frac{4}{5 + \frac{6}{7 + ...}}}$$ EDIT: Since the above has been determined as $\frac{1}{\sqrt{e}-1}$, is there a ...
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### Simpler closed form for $\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}$

I'm trying to find a closed form of this sum: $$S=\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}.\tag{1}$$ WolframAlpha gives a large expressions containing multiple ...
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Evaluate: $$\large\sum_{k=1}^{\infty}\left(\zeta{(2)}-\sum_{n=1}^{k}\frac{1}{n^2}\right)^2$$ MY ATTEMPT: Recognizing that $\zeta{(2)}-\sum_{n=1}^{k}\frac{1}{n^2}$ can be written as $... 4answers 105 views ### How to derive the closed form of the sum of$kr^k$$$\sum_{k=0}^{n}kr^k = r\frac{1-(n+1)r^n + nr^{n+1}}{ (1 - r)^2 }$$ How to derive it? I read about some finite calculus, and i understand how to tackle sums of$x^2$,$x^3$, etc.. But I don't know ... 3answers 49 views ### How to find a closed form of this simple factorial sequence [duplicate] $$S_1=1\\ S_n=n!+S_{n-1}$$ Is there a simple way to express$S_n$without summing up all the previous terms? Sorry I haven't put any effort in the problem but I don't know where to start. So this ... 4answers 405 views ### A sum containing harmonic numbers$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n}$I'm trying to find a closed form for the following sum $$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$ where$H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$is a harmonic number. Could you help me with it? 2answers 175 views ### Evaluation of a dilogarithmic integral Problem. Prove that the following dilogarithmic integral has the indicated value: $$\int_{0}^{1}\mathrm{d}x ... 1answer 81 views ### Compute the following series \sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)} Does the following series have a 'closed' form :$$\sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)}.$$Where n\in \Bbb{N} and a,b \in (0,+\infty) For a,b integer we can use Partial fraction ... 0answers 122 views ### A closed form for \sum_{k=1}^\infty \psi^{(1)} (k+a)\psi^{(1)} (k+b) The following result$$ \sum_{k=1}^\infty\left(\psi^{(1)} (k)\right)^2 = 3\zeta(3) $$where \psi^{(1)} is the polygamma function makes me think there is a nice sum for the series$$ ... 1answer 74 views ### Integral/infinite sum related to Bessels which pop up in optical coherence theory In propagating partially coherent optical fields, the following integral pops up: $$I_1=\int_0^{2\pi} e^{i(a\cos[\theta]+b\cos^2[\theta])}d\theta,$$ where$a$and$b$are real numbers. If we ... 1answer 105 views ### Evaluating an infinite square root How do I evaluate the square root: $$\sqrt{2013+276\sqrt{2027+278\sqrt{2041+280\cdots}}}$$ I have tried creating two arithmetic sequences such that $$a_n = 1999+14n$$ $$b_n = 274+2n$$ so the square ... 1answer 39 views ### Do all series have a closed form representation of their partial sum? If not, can we feasibly prove that this is not the case? The question was motivated by the way in which we approach the convergence and divergence of some series. During my undergraduate analysis course one of the only times in which the partial sum was ... 1answer 47 views ### Closed form for the recursion$\displaystyle u_n=\sum_{k=0}^{n-1} u_ku_{n-1-k}$I was completing a computer science problem when the following recursion popped up:$u_0=1\displaystyle u_n=\sum_{k=0}^{n-1} u_ku_{n-1-k}$Is there a closed form for this recursion ? I ... 2answers 60 views ### Find a closed form for the generating function for this sequence The sequence:$0, 0, 0, 1, 1, 1, 1, 1, 1, \ldots$The book gives the answer of$\frac{x^3}{1-x}$but I'm not sure how to get this answer. I understand the generating function of this sequence will be ... 1answer 56 views ### Find a sequence Find the function for the sequence$a_0 = 0, a_1 = 1$and$a_{n}=a_{n+10}+a_n$for all$n>0$. 3answers 187 views ### What is the closed form for$\sum_{n=1}^\infty \frac1n - \frac1{n+1/p}$? A while ago, I started to look at expressions of the following form: $$S_p:=\sum_{n=1}^\infty \frac1n - \frac1{n+1/p},$$ where$p$is prime, because otherwise things get too complicated for me at ... 1answer 243 views ### Approximate value of a slowly-converging sum of$\sum|\sin n|^n/n$In this question on Math.SE there appears this sum: $$S = \sum_{n\geq1}s_n, \qquad s_n = \frac{|\sin n|^n}{n},$$ which converges very slowly. What methods would you suggest for evaluating it ... 0answers 93 views ### What is the sum of Psi/Digamma-function of consecutive arguments? Is there a closed form? In a consideration of summation of a series $$s = a_0 + a_1 + a_2 + \cdots \tag 1$$ with $$\lim_{k \to \infty} a_k=0$$ but slowly decreasing, the coefficients$a_k$are somehow related to$1/k^2\$ ...
Let: $$\sum\limits_{k = 0}^n {k\left( {\matrix{ n \cr k \cr } } \right)} \cdot {4^{k - 1}} \cdot {3^{n - k}}$$ Find a closed formula (without summation). I think I should define this as a ...