Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

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Generalized Holder Inequality

Let $a_i \in \mathbb R^n$ with $a_i = (a_{i}^j)_{j = 1 ... n} = (a_{i}^1, ... ,a_{i}^n)$ for $i = 1, ... , k$ and let $p_1,...,p_k \in \mathbb R_{>1}$ with $\frac1{p_1}+ ... + \frac1{p_k} = 1$ ...
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0answers
59 views

Evaluation of a finite sum

I am having trouble evaluating the following finite sum: $$ \sum_{l=0}^{r}\binom{r}{l}(r-l)^{k},\qquad k\in\mathbb{N}_{0}. $$ Can you shed light on it?
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0answers
24 views

Can this infinite summation be simplified?

I encountered the following infinite summation $$\sum_{k=0,k\neq m}^{\infty}\frac{x^k}{(k-m)k!},x>0,$$ can it be simplified? Thanks!
3
votes
3answers
222 views

Is there a closed-form formula for sum of “odd combinations”? [on hold]

So, I was trying to come with a formula for the sum of below series: ${2^n \choose 1}+{2^n \choose 3}+...+{2^n \choose 2^n - 1}$ Thank you.
4
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2answers
59 views

Easier way to solve $\int_0^1 \frac{dx}{\lfloor{}1-\log_2(x)\rfloor}$

This problem showed up in the MIT integration bee last year: $$\int_0^1 \frac{dx}{\lfloor{}1-\log_2(x)\rfloor}$$ Basically, after doing a lot of tedious work I graphed out part of the function and ...
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0answers
20 views

Another sum involving binomial coefficients.

Let $a$ and $\theta$ be both real numbers not equal to a negative integer. Let $n$ and $m$ be positive integers. I have shown that the following equality holds: \begin{eqnarray} ...
4
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2answers
19 views

Evaluating $\sum_{i=a+1}^{N}\frac{i(i-1)}{i-a}$

I am trying to solve the German Tank Problem. There might be numerous ways of finding the expected value of N. However, the way in which I am proceeding, I need to find this sum. However I am stuck at ...
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2answers
80 views

$ \lim_{n \to \infty} \int_0^{\frac{\pi}{2}} \sum_{k=1}^{n} \left( \frac {\sin kx}{k} \right)^2 \, \mathrm{d}x $

Here is a problem in calculus shared by a friend. Compute $$ \lim_{n \to \infty} \displaystyle\int_{0}^{\frac{\pi}{2}} \displaystyle\sum_{k=1}^{n} \left( \frac {\sin kx}{k} \right)^2 \, \mathrm{d}x. ...
4
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1answer
44 views

Simplifying $\sum_{j=k}^{n}\binom{j}{k}/(2^{k-1})$

While doing an exercise (computing an expected value), I encountered an expression that looks like this. Is there a simpler formula? $$ \sum_{j=k}^{n}\frac{\binom{j}{k}}{2^{k-1}} $$ If it wasn't ...
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0answers
39 views

Evaluating $\sum \limits_{i=1}^{\infty} i^2 \exp\left[- \frac{(i+1/2)^2}{2s^2}\right] , \ s>0 $

How can we evaluate the following sum. $$ \sum \limits_{i=1}^{\infty} i^2 \exp\left[- \frac{(i+1/2)^2}{2s^2}\right] , \quad s>0 $$
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1answer
69 views

Can I demonstrate that this final sum is natural number? [on hold]

Let $m$ be some natural number. How can i say that $\sum _{k=0}^m\:\frac{1}{k!}$ is also natural or at least complete number?
3
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1answer
48 views

Alternating infinite sum

I have the following infinite sum: $$ \sum\limits_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{n}} $$ Because there is a $(-1)^n$ I deduce that it is a alternating series. Therefore I use the alternating series ...
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0answers
25 views

Existence and uniqueness of a function generalizing a finite sum of powers of logarithms

I hope to find a proof of the following conjecture: $(1)$ For every $a>0$ there is a convex analytic function $f_a:\mathbb R^+\to\mathbb R$ such that: $f(1)=0$ and $\forall x>1,\ ...
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1answer
16 views

Random Variable probability summation tweaking

I can't seem to figure out what they do to get to the bottom
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1answer
49 views

Sum of logarithmic series

Let $x_1>0$ we define sequence $(x_n)$ with formula $x_{n+1}=-\ln(x_1+x_2+\cdots+x_n)$ Find sum of the series $\sum_{n=1}^\infty x_n$. How to deal which such summation with logarithms?
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2answers
11 views

Convergence depending on the parameter

Let $c\ge 0$ be a real number. Then we define $$a_1=1, \quad a_{n+1}=\frac{cn+1}{n+3} a_n$$ Investigate convergence of $\displaystyle \sum_{n=1}^{\infty} a_n$ depending on the parameter $c$. Here I ...
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1answer
34 views

Convergence of a series and sum [on hold]

I have the followin series: $\sum_{n=2}^\infty\left(\sum_{k=2}^n(-2)^{-k}3^{-n+k}\frac{1}{k!}\right)$ I need to investigate convergence of the series and calculate its sum. How to do this (both)? In ...
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0answers
84 views

How to sum up this series and simplify yet another one?

Primarily, I would like to know what could be done wit this series: $$ \sum_{n=2}^{\infty}\frac{n^3}{(n^2-1)^3}\left(\frac{n-1}{n+1}\right)^{2n}$$ Moreover, I would like to simplify the following ...
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4answers
55 views

How many possible 1mb files are there? [on hold]

If you look at all combinations of data that can be stored in a 1mb file, how many are there before you have every possible 1mb file? How much space does that take up?
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2answers
55 views

How to change the limits of a summation when the index $k$ is replaced by $-k$?

Is what I am doing below correct, please assist. $$\sum_{k=-\infty}^{-1}\frac{e^{kt}}{1-kt} = \sum_{k=1}^{\infty}\frac{e^{-{kt}}}{1-kt}$$ Is this the rule on how to "invert" the limits, and does ...
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2answers
32 views

How to simplify this summation: $\frac{2\sum_{k=0}^n2^n}{n+1}=2^{n+1}$?

So I saw an earlier post where they had this equation here. $\frac{2\sum_{k=0}^n2^n}{n+1}=2^{n+1}$? However, I do not know how they did this? Am I missing something?
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2answers
49 views

Prove that $a_n = 2^n$

Let the recurrence relation $$ a_0 = 1 \\ a_{n+1} = \frac{2 \sum_{k=0}^n a_ka_{n-k}}{n+1} $$ I need to find a close formula for this recurrence. I've noticed that $a_n = 2^n$. I tried to prove it ...
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1answer
50 views

Given $f'(x) = 2f(x)^2$ find the recurrence formula for $a_n$ in $f(x) =\sum a_n x^n$.

Let $f(x) = \sum_{n=0}^\infty a_n x^n$ with radius convergence of $R>0$. We know that $f'(x) = 2f(x)^2$. Find the recurrence formula of $a_n$. I don't know if it makes a difference but $f(x)$ ...
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2answers
41 views

How to calculate the following sum?

$\sum _{n=1}^{\infty \:}\frac{10000}{\left(2n+3\right)^4}$ I could only prove that it is convergent, but I have no idea how to find the sum. Thanks for the help :-)
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3answers
42 views

Proving a formula with binomial coefficient

Is this formula true? How can I prove it? $$\sum_{s=0}^{n-1}\binom{n-1}{s}2s =2^{n-1}(n-1)$$ Thanks!
2
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1answer
58 views

Simplify a sum of binomial

Is it possible to have a closed form of the following sum: $$\sum_{i=0}^n\binom{n}{i}\binom{n+t-i}{n}$$
10
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2answers
236 views

$\pi$ in terms of $4$?

I'm trying to define $\pi$ in terms of $4$ by placing a unit circle inside a square, and subtracting the corners of the square. I'm attempting to use summation to define the area of a corner, then ...
3
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1answer
55 views

A “contradiction” to Abel's theorem for series

I proved that for every $0<a<1$, the series $ \sum_{n=0}^\infty \left( \frac{x^{2n+1}}{2n+1} - \frac{x^{n+1}}{2n+2} \right) $ converges uniformly at $[-a,a]$ to $\frac{1}{2}\log (x+1)$. Now, ...
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1answer
67 views

How to calculate the sum of an infinite series [duplicate]

How do you calculate the sum of an infinite series like $$ \sum_{n = 0}^\infty \frac{n}{2^\sqrt{n}}$$ //EDIT //Ignore I searched up how to find this with infinite geometric series solution which ...
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2answers
45 views

Please help prove this summation problem for me [on hold]

Prove that for all integers n greater than or equal to 1, $\sum_{k=1}^{3n} (4k+3)=3n(6n+5)$.
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3answers
423 views

A Sum that came up while solving a integral

While evaluating $I$, I did the following- $$\begin{align}I= \int_{0}^{1} \log \left(\dfrac{1+x}{1-x}\right) \dfrac{1}{x\sqrt{1-x^2}} \ \mathrm{d}x &= 2 \int_{0}^{1}\sum_{n=0}^{\infty} ...
2
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1answer
24 views

Is it possible to find the limit of this?

After recently watching a Numberphile video about a square problem I started thinking about what would happen to the sum of all angles if you had n amount of squares. After a bit of testing, I ...
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3answers
39 views

convergence of infinite series $\sum_{n=1}^\infty \frac{x^n}{(1+x)(1+x^2)(1+x^3)\cdot\cdot\cdot (1+ x^n)}$

I am reviewing for my final exam, and viewed this question: Decide whether the following infinite sum is convergent for all $x >1$: $$\sum_{n=1}^\infty ...
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2answers
58 views

Prove that $\sum_{n=0}^\infty \frac{(-1)^n}{3n+1} = \frac{\pi}{3\sqrt{3}}+\frac{\log 2}{3}$

Prove that $$\sum_{n=0}^\infty \frac{(-1)^n}{3n+1} = \frac{\pi}{3\sqrt{3}}+\frac{\log 2}{3}$$ I tried to look at $$ f_n(x) = \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^n $$ And maybe taking it's ...
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3answers
18 views

Sum of segments inside a right triangle.

I am interested for a problem involving the sum of segments inside a right triangle. Consider a right triangle of hypotenuse $\overline{BC}$ and catheti $\overline{AB}$ and $\overline{AC}$. From the ...
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1answer
46 views

Is there a summation formula for this equation (contains square roots, and functions within the square root)?

I am trying to solve a summation formula that is quite complex. However, to make the "answering" process for you guys easier I'll isolate the part I am having trouble with... The equation is as ...
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0answers
42 views

Other Patterns in Triples

I have the following 20 triples generated by polynomial distribution: $$\begin{matrix} (2,4,5)&(2,3,4)&(2,3,5)&(1,4,5)&(2,2,4)\\ ...
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2answers
52 views

Convolution of discrete uniform distributions

For two independent, discrete, uniformly distributed random variables $X$ and $Y$, I wish to obtain the distribution of the sum $Z=X+Y$. I have the densities: $$f_X(x)=\left\{\begin{matrix} ...
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1answer
78 views

How to prove this Catalan number identity

Catalan number is $\displaystyle C_n= \frac{1}{n+1}\binom{2n}{n}$. How to prove that $$C_{2n-1} = \sum_{k=0}^{n-1}\left(\binom{2n-1}{n-k-1}-\binom{2n-1}{n-k-2}\right)^2$$ for $n\geq 1$. Thank you.
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1answer
34 views

Estimation of a sum independent of $n$

Suppose $f$ is differentiable on $[0,1]$, $f(0)=f(1)$, $\int_0^1 f(x)dx=0$, $f'(x)\neq 1$. Furthermore, let $g(x)=f(x)-x$, $n\geq 2$ is an integer. Show that ...
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4answers
96 views

Does $\sum_{n=1}^{\infty}\frac{\cos\left(\frac{n\pi}{2}\right)}{\sqrt{n}}$ converge?

Does the following series converge? $$\sum_{n=1}^{\infty}\frac{\cos\left({\frac{n\pi}{2}}\right)}{\sqrt{n}}$$ The $\cos$ function: alternates between (-1) and 1 for every $n$ that is even. ...
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1answer
25 views

Product of sums equal to sum of products

Is $(\sum_k x_k)(\sum_i y_i)(\sum_j z_j)(\sum_l a_l) = \sum_{kijl}x_ky_iz_ja_l$ with $\sum_{kijl} = \sum_k\sum_i\sum_j\sum_l$?
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3answers
120 views

How to prove combinatorial identity $\sum_{k=0}^s{s\choose k}{m\choose k}{k\choose m-s}={2s\choose s}{s\choose m-s}$?

The following combinatorial identity have been verified via maple, but I can not prove it. Who can prove it without WZ mehtod? $$\sum_{k=0}^s{s\choose k}{m\choose k}{k\choose m-s}={2s\choose ...
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2answers
102 views

How prove this identity$\sum_{k=0}^{n}\binom{2k}{k}\binom{n+k}{2k}(s-t)^{n-k}t^k=\sum_{k=0}^{n}\binom{n}{k}^2s^{n-k}t^k$

Today I see a paper,and this author say it is easy to have this identity.But I take sometimes to prove it,and I can't prove it. show this following identity holds for any real $s$ and $t$ and any ...
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3answers
73 views

Prove that $1+2^1+2^2+\ldots +2^n=2^{n+1}-1$ using induction

For all integers $n\ge 1$ prove the following statement using mathematical induction. $$1+2^1+2^2+\ldots +2^n=2^{n+1}-1$$ The first part of the question ask me to prove the base step: So I set ...
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4answers
75 views

How to sum $\sum_{n=1}^{\infty} \frac{1}{n^2 + a^2}$?

Does anyone know the general strategy for summing a series of the form: $$\sum_{n=1}^{\infty} \frac{1}{n^2 + a^2},$$ where $a$ is a positive integer? Any hints or ideas would be great!
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2answers
24 views

Generating functions and central binomial coefficient

How would you prove that the generating function of $\binom{2n}{n}$ is $\frac{1}{\sqrt{1-4y}}$? More precisely, prove that( for $|x|<\frac{1}{4}$ ): ...
0
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1answer
26 views

Determine the convergence of the following series

$$\sum_{k=0}^\infty {3^{k\ln k} \over {k^k}}$$ I need to determine the convergence of this series. I know it diverges, but how do I prove this?
0
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0answers
38 views

What is the sum of a finite series when the variable is in the denominator?

I would like to know if there is a closed form way of writing this sum please: $$\sum_{i=1}^n\frac{a}{b-x_i}$$
1
vote
1answer
31 views

Summation with two running indices

I don't understand the notation of the following summation. $$ \sum_{i,j=1}^m \gamma_i \cdot \beta_{ij} \cdot \alpha_j$$ I first thought $ i, j $ would be increased simultaneously, but that would ...