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

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5
votes
1answer
69 views

Summation of factorials.

How do I go about summing this : $$\sum_{r=1}^{n}r\cdot (r+1)!$$ I know how to sum up $r\cdot r!$ But I am not able to do a similar thing with this.
4
votes
0answers
64 views

Dealing with a difficult sum of binomial coefficients, $\sum_{l=0}^{n}\binom{n}{l}^{2}\sum_{j=0}^{2l-n}\binom{l}{j} $

I am interested in finding an upper bound for the sum $$F(n)= \sum_{l=0}^{n}\binom{n}{l}^{2}\sum_{j=0}^{2l-n}\binom{l}{j}.$$ Ideally it should be possible to evaluate it exactly using some ...
1
vote
0answers
30 views

Calculate the limit of $\frac{1}{n+1} + \frac{1}{n+2} + … + \frac{1}{2n}$ [duplicate]

I'm stucked at calculation this limit. Using $Z_n =\sum\limits_{n=1}^{\infty} \frac{1}{n}$ I could rewrite my series as $Z_{2n} - Z_n$ but I couldn't go any further since harmonic series diverges and ...
1
vote
2answers
42 views

Sum of elements of a recursive sequence

Say we have $$a_n=\frac{n-1+b}{n-b}a_{n-1}$$ with $a_0>0$ and $-1<b<0$. Then it holds that $$\sum_{n=1}^{\infty}a_n=-\frac12 a_0.$$ I could establish this using simple manupulations with ...
4
votes
4answers
96 views

How to calculate $\int_a^bx^2 dx$ using summation?

So for this case, we divide it to $n$ partitions and so the width of each partition is $\frac{b-a}{n}$ and the height is $f(x)$. \begin{align} x_0&=a\\ x_1&=a+\frac{b-a}{n}\\ &\ldots\\ ...
0
votes
3answers
75 views

limit of summation $\lim_{N \to \infty}\sum\limits_{n=1}^{N} \frac{n^2+1}{2^n}$

I was wondering if anyone could help me with this limit. I tried to separate the sums into $\lim_{N \to \infty}\sum\limits_{n=1}^{N} \frac{n^2}{2^n}$ which equals to 0 and $\lim_{N \to ...
0
votes
1answer
14 views

Ratio of sums vs sum of ratio

Is anyone aware of any general (or perhaps not so general) relationship (inequality for instance) relating $A(x,y)= \frac{\sum_z f(x,y,z)}{\sum_z g(y,z)}$ and $B(x,y)= ...
1
vote
0answers
38 views

Is there a closed form expression for the following sum?

Is there a closed form expression for the following sum? $$\sum_{0\le i_1<i_2<\cdots<i_k\le n}r^{i_1+i_2+\cdots+i_k}$$ I can understand that there are $\binom{n}{k}$ such terms and the ...
2
votes
2answers
53 views

How can I tell which one of these numbers is greater?

I have two very large numbers how do I tell which one is greater. The two numbers are $$\sum^{9(10^{99}) } _{i=1}i^9$$ and $9^{9^{9^{9^{9^{9^{9^{9^{9^{9}}}}}}}}}$
0
votes
0answers
6 views

Discretization of an integral

I have a question about integral discretization - maybe it is a bit silly. This is my starting point: $$ f(a)=\int_{20}^{a}e^{(a-s)}e^{-\int_{20}^{s}\lambda(u)du}\lambda(s)ds $$ In practice, a is ...
1
vote
1answer
56 views

Question on a step in a proof regarding complex roots of unity powers

Given that I have the expression $$N^{k-1}k!\sum_{n=0}^{N-1}w_N^{-nk}e^{\frac{w_N^{n+m}x}{N}}$$ and given that $N=2m$, The next line of the proof states that the above is equal to ...
1
vote
2answers
37 views

Is there a name for this double summation identity? What is the shortest way to illustrate that it holds?

Say I have the following the expression: $$\sum\limits_{j=0}^{i-1} \sum\limits_{u=0}^{j} g(u)$$ By enumeration, it is easy to see that: in the case when $j=0$ we have $$\sum\limits_{u=0}^{j} g(u) ...
1
vote
1answer
39 views

Show that $\sum_{i=\lceil\alpha n\rceil}^n {n\choose i}\le 2^{nH(\alpha)}, \alpha\in(1/2,1]$

For $1/2<\alpha\le 1$ show that $$\sum_{i=\lceil\alpha n\rceil}^n {n\choose i}\le 2^{nH(\alpha)}$$ where $H(\alpha)=-\alpha\log_{2}\alpha - (1-\alpha)\log_2 (1-\alpha)$ is the entropy. ...
0
votes
1answer
21 views

Finding $\mathbb{E}[Y^2]$ (Negative Binomial Distribution)

Suppose $Y \sim \text{NegBin}(n, p)$. Then $$\mathbb{E}[Y^2] = \sum_{y=0}^{\infty}y^2\binom{r+y-1}{y}p^rq^y$$ where $q = 1 - p$. Now since the $y = 0$ term adds nothing, $$\begin{align} ...
1
vote
0answers
38 views

How do you express $\sum_{j=1}^{n} j^{k+1}$ in terms of $\sum_{j=1}^{n} j^{k}$?

I am trying to use induction to prove, for every positive integer $k$, that $$\sum_{j=1}^{n} {j^{k}} = \frac{n^{k+1}}{k+1} +\frac{n^k}{2} + P_{k-1}(n)$$ where $P_{k-1}$ is a polynomial of degree at ...
1
vote
0answers
20 views

The sum of $1/(ci-1)$ from $i=n\alpha$ to $n-1$

For fixed $c>0$ and $\alpha \in (0,1)$, find the sum (as $n \to \infty$) of: $$ \sum_{i=n\alpha}^{n-1} \frac{1}{ci-1} $$ This looks like a harmonic sum which doesn't converge, but the fact that it ...
4
votes
3answers
86 views

Prove that $\sum_{k=0}^m \dbinom{n}{k} \dbinom{n-k}{m-k}= 2^m \dbinom{n}{m}$ for $m < n $

Prove that $\sum_{k=0}^m \dbinom{n}{k} \dbinom{n-k}{m-k}= 2^m \dbinom{n}{m}$ for $m < n $ In short : I've tried to prove this by induction since I can't really see how to interpret this with a ...
0
votes
1answer
29 views

Find the conditional distribution of several variables given their sum

The question is: Let $Y_1, Y_2, ..., Y_k$ be $k$ independent Poisson random variables with parameters $\lambda_1, \lambda_2, ..., \lambda_k$ respectively. Find the conditional distribution of ...
1
vote
2answers
98 views

To evaluate the sum $\frac{1}{5}-\frac{1 \cdot 4}{5 \cdot 10}+\frac{1 \cdot 4 \cdot 7}{5 \cdot 10 \cdot 15}-\ldots$

Right now I am working through archived papers of a math aptitude quiz. For some reason I seem to be haveing a hard time with these series problems. I have managed to write the above series in a ...
1
vote
1answer
35 views

The sequence $x_1,x_2,x_3,\cdots$ is defined by $x_1=2$ and $x_{k+1}=x_k^2-x_k+1$ for all $k \ge 1$. Find $\sum_{k=1}^\infty \cfrac{1}{x_k} $

The sequence $x_1,x_2,x_3,\cdots$ is defined by $x_1=2$ and $x_{k+1}=x_k^2-x_k+1$ for all $k \ge 1$. Find $\sum_{k=1}^\infty \cfrac{1}{x_k} $ By experimenting ,I was able to prove by ...
2
votes
1answer
83 views

How can we prove the sum of squares/cubes/etc is always a polynomial of appropriate degree?

What elementary proofs are there that $$\sum_{k=1}^{n}k^m$$ is always a degree $m+1$ polynomial? It is well known that $$\sum_{k=1}^{n}k^2=\frac{n(n+1)(2n+1)}{6}$$ I'm interested in sums ...
0
votes
1answer
55 views

Why does this partial derivative of a summation work?

I'm trying to take the partial derivative of $-\sum\limits_{i=1}^n \frac{(x_i-\mu)^2}{2\sigma^2}$ with respect to $\mu$. The correct answer is $\sum\limits_{i=1}^n \frac{x_i-\mu}{\sigma^2}$. It ...
2
votes
3answers
48 views

Solution for $\sum_{i=0}^{n} \sin(\frac{i\pi}{2n})$? [duplicate]

While I was trying to find the formula of something by my own means I came across this sum which I need to solve, however I don't know if there is a solution for it, maybe it doesn't mean anything and ...
0
votes
0answers
20 views

sum of fractional order Legendre functions

I need to calculate the sum \begin{equation} \sum_{l=0}^{\infty}{\Gamma(l+N) \over \Gamma(l+1)}{\left[\Gamma\left({l \over N}+1\right)\right]^2 \over \Gamma\left(2{l \over N}+1\right)}P_{l\over N}(z) ...
0
votes
0answers
41 views

How is $\sum_{i=0}^{\log n -1} 2^i(\log n -i) =\sum_{i=1}^{\log n} i\frac n {2^i} $

I found the following in a textbook: $$\sum_{i=0}^{\log n -1} 2^i(\log n -i) =\sum_{i=1}^{\log n} i\frac n {2^i} $$ It's a summation of a chart, the explanation for this was to "flip the chart", I ...
1
vote
0answers
24 views

Proof of sine integral

How to prove: $$\operatorname{Si}(x) =\sum_{k=0}^\infty { \frac { -\sin\left( \frac{\pi k}{2} \right) }{ k\times k! } {(-x)}^k} $$ Here $\operatorname{Si}(x)$ is Sine Integral. Actually I ...
1
vote
1answer
48 views

Order of an infinite sum

How to prove that, for $0<c<1$, $$ \sum\limits_{j=1}^{\infty} c^{\, j } \cdot j^{\, -(\frac{d}{2} +1 )} $$ is, for some positive constant $K$, of order $K + O( \, ( 1-c )^{\frac{d-2}{2}})$ when ...
1
vote
3answers
55 views

prove that $ \binom{n-1}{0} +\binom{n}{1}+\binom{n+1}{2}+\cdots+\binom{n+k}{k+1}=\binom{n+k+1}{k+1}$

I am asked to prove that $$ \dbinom{n-1}{0} +\dbinom{n}{1}+\dbinom{n+1}{2}+\cdots+\dbinom{n+k}{k+1}=\dbinom{n+k+1}{k+1}$$ So far what I've tried ,without looking to much at the sum I've to prove ,is ...
0
votes
1answer
45 views

Can't understand this step in Zeta-function transformations

Sorry for exceptional dullness, but I can't understand this step in Zeta-function transformations: $$ζ(s)= \sum_{n=1}^\infty \frac{1}{n^s}= \sum_{n=1}^\infty n (\frac{1}{n^s}-\frac{1}{(n+1)^s})$$
2
votes
1answer
50 views

Calculate $\sum_{k=0}^\frac{n}{2}(-1)^k{n \choose 2k}$ and $\sum_{k=0}^\frac{n-1}{2}(-1)^k{n \choose 2k+1}$

a) $\sum_{k=0}^\frac{n}{2}(-1)^k{n \choose 2k}$ b) $\sum_{k=0}^\frac{n-1}{2}(-1)^k{n \choose 2k+1}$ I think a way to calculate the sums is to see what happens to $(1+i)^n$ but after trying for 2 ...
-2
votes
1answer
33 views

Convergence of infinite sum including cosh functions?

I am attempting to code up an equation that includes an infinite sum of cosine and hyperbolic cosine functions, namely: $$ \sum_{m=0}^{\infty} \frac{ \cos[(2m+1)\pi x/s] \cosh[(2m+1)\pi z/s] } ...
4
votes
3answers
90 views

mysterious sum of two sequences

Let $$S_1 = \sum_{n=1}^\infty \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \cdots$$ $$S_2 = \sum_{n=1}^\infty ...
0
votes
2answers
42 views

Evaluate the sum $\sum_{k=0}^n 2^k \cdot \dbinom{n}{k}$

Evaluate the sum $\sum_{k=0}^n 2^k \cdot \dbinom{n}{k}$ I think the problem calls for some application of Vandermonde's identity as the author previously proved this identity,however I can't see ...
-1
votes
0answers
53 views

Lost proof of trigonometric formula

The following formula seems to be true for odd positive integers $n$ but i forgot the way I proved it $$\sum_{k=1}^n\tan(\alpha+\frac{k2\pi}{n})=n\tan(n\alpha)$$ Maybe someone can deliver the ...
2
votes
2answers
71 views

Closed form or approximation of $\sum\limits_{i=0}^{n-1}\sum\limits_{j=i + 1}^{n-1} \frac{i + j + 2}{(i + 1)(j+1)} (i + 2x)(j +2x)$

During the solution of my programming problem I ended up with the following double sum: $$\sum_{i=0}^{n-1}\sum_{j=i + 1}^{n-1} \frac{i + j + 2}{(i + 1)(j+1)}\cdot (i + 2x)(j +2x)$$ where $x$ is some ...
-1
votes
3answers
89 views

Proving that $\sum_{j=0}^n(-1)^j\binom{n}{j} = \binom{n}{0} - \binom{n}{1} + … \pm \binom{n}{n}=0$ [duplicate]

The equation to be proved is: $\sum_{j=0}^n(-1)^j\dbinom{n}{j} = \dbinom{n}{0} - \dbinom{n}{1} + ... \pm \dbinom{n}{n}=0$ But if i take the base case ($n = 1$) i get $\sum_{j=0}^n(-1)^j\dbinom{n}{j} ...
4
votes
3answers
107 views

Induction on inequalities (a sum less than a particular value) [duplicate]

I am trying to solve this inequality by induction. I just started to learn induction this week and all the inequalities we had been solved were like an equation less than another equation (e.g. $n! ...
-1
votes
1answer
61 views

The name of the sum $\sum_{i=0}^n \frac{1}{m-i}$

Sorry for the vague question name, since I am looking for the name of the series. Also this might not be a "series" by the strict definition of a series.. anyways here it is: Choose some $m$ and $n$ ...
2
votes
2answers
56 views

Induction proof without summation

I have to prove this induction: $\dfrac{1}{(n+1)}+\dfrac{1}{(n+2)}+\dots+\dfrac{1}{2n} = \dfrac{1}{(1\times2)}+\dfrac{1}{(3\times4)}+\dots+\dfrac{1}{(2n-1)\times2n}$ Can someone help me with it?
0
votes
0answers
34 views

come up with a formula for $f(n)=\sum_{i=1}^n i^k$ [duplicate]

I know that the general method is to attempt getting the constant out of the summation, then use the summation formula on that and multiply the two. Is there any way to get K outside of the summation ...
-1
votes
3answers
75 views

On the sum of the reciprocals of square roots. [closed]

What is the analytic sum of $1+ \frac{1}{\sqrt 2} + \cdots + \frac{1}{\sqrt x}$ ?
1
vote
1answer
40 views

is there a difference between sum and integral?

is there a difference between integrating a function between two limits and summing a function and if so where does the difference come from and when would you use each method in real life situations
0
votes
0answers
43 views

Simplify sum with binomials

An algorithm finds prefixes of given length k from given word with length n. It is required to find the time complexity of given algorithm. It is easy when no nodes get cut off in its recursion tree ...
0
votes
0answers
35 views

How to calculate this infinite sum [duplicate]

I don't understand how to get from the infinite sum $\sum_\limits{n=1}^{\infty}\frac{1}{n^2}$ to the solution $\frac{\pi ^2}{6} $. Is it possible to calculate this by hand or do I just have to type ...
3
votes
3answers
95 views

Proof that $\sum_{i=0}^n {n\choose i}2^{i}=3^{n}$ [duplicate]

The following sum came up in a combinatorial argument. I know what it equals thanks to Wolfram Alpha, but I'm not sure how to show it $$\sum_{i=0}^n {n\choose i}2^{i}=3^{n}$$
2
votes
2answers
59 views

Writing sigma notation $\sum^n_{i=1} \frac {i}{2^i}$ in closed form

What would be a way to find the closed form of $\frac {1}{2} + \frac {2}{4}+\frac {3}{8}+\cdots+\frac {n}{2^n}=\sum^n_{i=1} \frac {i}{2^i}=s$ I've looked at $\frac {s}{2}=\frac {1}{4} + \frac ...
0
votes
2answers
29 views

Sum of first $n$ positive integers to a positive power $p$

Consider the sum $$\sum_{i=1}^{n}i^{p}\text{ , }p \in \mathbb{Z}^+\text{.}$$ Using a method in Spivak's Calculus, it can be shown that $$(n+1)^{p+1}-1 = ...
1
vote
3answers
56 views

Converting $(1+…+n)^2*(n+1)^3$ to $(2+…+2n)^2$

I'm currently going through Calculus by Spivak by myself, and came across a proof by induction requiring to prove $1^3+...+n^3 = (1+...+n)^2$ Naturally, to prove this, I need to somehow convert ...
0
votes
2answers
39 views

Is $\sum_{i=1}^{n}\frac{f(c\frac{i}{n})}{f(c\frac{i}{n})+f(c-c\frac{i}{n})}=\frac{1}{2}(n+1)$?

So I was trying some code on Octave. The algorithm is the following $$\sum_{i=1}^{n}\frac{f(c\frac{i}{n})}{f(c\frac{i}{n})+f(c-c\frac{i}{n})}$$ for some $n\in\mathbb{N}$ and $c\in\mathbb{R}$. I ...
1
vote
1answer
40 views

How do you find the sum: $\sum_{r=1}^6 \tan^2\left(\frac{r \pi}{n}\right)$

How do you find the sum: $$\sum_{r=1}^6 \tan^2\left(\frac{r \pi}{n}\right)$$ I managed to solve this question using complex numbers so I thought I'd share the solution. If you know of any better ...