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

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0
votes
1answer
28 views

Quicksort-How did we get the relation?

At the proof of the theorem that the expected time of Quicksort is $O(n \log n)$, there is the following sentence: We suppose that the partitions are equally likely, so the possibility that the sizes ...
0
votes
0answers
16 views

Find a probability of $n$ event happening from $m$ types

The question is: to find a sum $$ S=\sum\limits_{n_1+n_2+\ldots+n_m = n,\ n_i=0,1,\ldots,n} p_1^{n_1}p_2^{n_2}\cdots p_m^{n_m}, $$ where $p_i\in[0,1]$. UPDATE. This issue has no probabalistic ...
1
vote
1answer
60 views

What is the probability of choosing r objects from c different groups when there are m groups of n objects?

Suppose I have m groups of n objects each for a total of nm objects. I am going to choose r of these nm objects. I want to know what the probability is that my r objects come from c different ...
2
votes
1answer
38 views

How find this sum closed form $I=\sum_{k=1}^{n}\int_{0}^{+\infty}\cos{(2kx)}x^{m-1}e^{-ax}dx$

Find this closed form? $$I=\sum_{k=1}^{n}\int_{0}^{+\infty}\cos{(2kx)}x^{m-1}e^{-ax}dx,m\ge 1,a>0$$ use ...
26
votes
6answers
3k views

Is it possible to write a sum as an integral to solve it?

I was wondering, for example, Can: $$ \sum_{n=1}^{\infty} \frac{1}{(3n-1)(3n+2)}$$ Be written as an Integral? To solve it. I am NOT talking about a method for using tricks with integrals. But ...
6
votes
4answers
146 views

How to compute $\sum_{n\ =\ 1}^{\infty}\arctan\left(\,\frac{3n^{2}}{ 2n^{4} - 1}\,\right)$

I find this problem on facebook group. $$\mbox{Is it possible to find exact value of}\quad \sum_{n\ =\ 1}^{\infty}\arctan\left(\,\frac{3n^{2}}{ 2n^{4} - 1}\,\right)\ {\large ?}. $$ I think this is ...
1
vote
1answer
41 views

An unusual two dimensional sum

Can anyone prove or reference a proof for the following bound (unless it's not true!) $$\sum_{|\underline{k}|_{\infty} > M} \frac{1}{((k_1)^2 + (k_2)^2 )^2} \leq \frac{C}{M^2}$$ where ...
0
votes
0answers
28 views

A multiple sum involving binomial coefficients.

Let $n\ge 1$ and $0 < a < b$ be integers and $\vec{p}:= \left(p_l\right)_{l=1}^q$ be a vector of real numbers. The question is to find the following sum. \begin{equation} {\mathfrak ...
0
votes
1answer
50 views

Can this binomial summation be simplified?

I got something like $\displaystyle\sum_{i=0}^K{ \binom{n+i}{i} \cdot \alpha^i} $ where $n,\ K,\ \alpha$ are definite values, $\binom{n+i}{i}$ is the Combinatorial number that choose $i$ from ...
2
votes
0answers
53 views

Prove that $\sum\limits_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2})$. [on hold]

Let $a,b\in\mathbb{Z}$, and $f\in C^2([a,b])$ such that $|f''(t)|\asymp \lambda$ for $a\le t\le b$. Prove that $$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}).$$ ...
1
vote
0answers
28 views

Sum of all n dimensional spheres?

I was messing around and made some code to find the area of an n dimensional sphere. I noticed that as n increases, the area tends towards zero. These were the results: ...
13
votes
2answers
298 views
+50

$\pi$ 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 ...
5
votes
1answer
46 views

How to prove that $f(x) - f(x-1)$ approaches $\frac{\log_{10}(10)}{\log_{10}(e)}$?

Let $$f(x) = \sum_{n=1}^{10^x}\frac{1}{n}$$ I noticed that as x approaches $\infty$, $f(x) - f(x - 1) \approx 2.3025$. After a bit of experimenting, I found that $2.3025... = ...
3
votes
3answers
95 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. ...
1
vote
1answer
49 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!
-1
votes
1answer
19 views

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 $\frac1{p_1}+ ... + \frac1{p_k} = 1$ ...
1
vote
0answers
34 views

A sum involving binomial coefficients and a simple fraction

Let $a_1$ and $a_2$ be real numbers. Let $n_1$ and $n_2$ be positive integers. Finally let $\theta$ be a real number which is different from a negative integer. By the generalizing the result from ...
4
votes
2answers
54 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 ...
0
votes
4answers
44 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!
1
vote
0answers
81 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?
3
votes
3answers
233 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
votes
2answers
61 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 ...
0
votes
0answers
26 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
votes
2answers
20 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 ...
4
votes
1answer
89 views

Why are infinite sums so much harder to calculate than the associated infinite integral?

It seems that with continuous functions, we have in calculus an apparatus for "short cutting" an infinite sum. However, when we move to the discrete case, it seems that we don't have the equivalent ...
2
votes
1answer
45 views

How to correctly represent a nested sum

Suppose I have a matrix: $$ A = \begin{pmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \\ \end{pmatrix} $$ For which i want to sum the ...
0
votes
1answer
361 views

Normalize sum of values to percentages from 0-100

I have two values that sum to one in an arbitrary manner: [.6 , .4] [-.2, 1.2] [1.9, -.9] etc... How can I "normalize" these values so that they represent a percentage from 0-100%? What ...
1
vote
0answers
42 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. $$ \sum \limits_{i=1}^{\infty} i^2 \exp\left[- \frac{(i+1/2)^2}{2s^2}\right] , \quad s>0 $$
1
vote
0answers
43 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)\\ ...
3
votes
3answers
140 views

Show that $\lim_{n\to\infty} \sum_{k=1}^{n} \frac{n}{n^2+k^2}=\frac{\pi}{4}$

Show that $$\lim_{n\to\infty} \sum\limits_{k=1}^{n} \frac{n}{n^2+k^2}=\frac{\pi}{4}$$ Using real analysis techniques.
1
vote
1answer
55 views

Looking for $\sum_{n=1}^{\infty}\frac{1}{F_{n}F_{n+2}}$

I am looking for the sum of the series: $$\sum_{n=1}^{\infty}\frac{1}{F_{n}F_{n+2}}$$ where $F_{n}$ is the $n$-th Fibonacci number. I was thinking about splitting the fraction into 2 like in the ...
0
votes
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?
4
votes
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 ...
1
vote
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 ...
3
votes
1answer
49 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 ...
1
vote
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,\ ...
0
votes
1answer
50 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?
0
votes
1answer
16 views

Random Variable probability summation tweaking

I can't seem to figure out what they do to get to the bottom
0
votes
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 ...
-1
votes
1answer
34 views

Convergence of a series and sum [closed]

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 ...
3
votes
2answers
2k views

Prove $\sum \binom nk 2^k = 3^n$ using the binomial theorem

I'm studying for a midterm and need some help with proving summation using the binomial theorem. $\sum\limits_{k=0}^n {n \choose k} 2^k = 3^n$ This is what I'm thinking so far: In the binomial ...
18
votes
1answer
599 views

How to prove a double sum is always an integer?

I have verified the following double sum is always an integer for $s$ up to $1000$ via Maple. But I can not prove it. Proofs, hints, or references are all welcome. Thanks! ...
2
votes
1answer
59 views

Simplify a sum of binomial

Is it possible to have a closed form of the following sum: $$\sum_{i=0}^n\binom{n}{i}\binom{n+t-i}{n}$$
-5
votes
4answers
57 views

How many possible 1mb files are there? [closed]

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?
6
votes
3answers
430 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} ...
0
votes
2answers
56 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 ...
0
votes
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?
0
votes
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 ...
1
vote
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 $f(x)$ ...
0
votes
2answers
42 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 :-)