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Questions tagged [summation]

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

0
votes
2answers
45 views

Closed form for $\sum_{n=1}^\infty \log(n) * x^n$

As in the title, I'm in quest for $\sum_{n=1}^\infty \log(n)\cdot x^n$, where $0 \le x \lt 1$ Wolfram Alpha says: $-\operatorname{PolyLog}^{(1, 0)}(0, x)$, but I don't understand what that means. (...
0
votes
1answer
23 views

Fredholm integral equation needs to be written as a sum of functions

solve the equation $$ f(x) + \int_0^1 (xy+x^2y^2) f(y) dy = g(x) $$ and write in the form of $$ \sum a_jx^{j-1} $$ I have tried integration by parts but it doesn't seem to work because of f(y). ...
1
vote
3answers
87 views

The series $\frac{1}{2}+\frac{2}{5}+\frac{3}{11}+\frac{4}{23}+…$

Consider the expression $\frac{1}{2}+\frac{2}{5}+\frac{3}{11}+\frac{4}{23}+...$ Denote the numerator and the denominator of the $j^\text{th}$ term by $N_{j}$ and $D_{j}$, respectively. Then, $N_1=1$, ...
0
votes
1answer
28 views

Proof for sum of digits of a number until sum is a single number

Here is a more elaborate description of the problem statement. What I found with a few examples is that given a number, say 569. If we are required to sum its digits repetitively until the sum is ...
1
vote
4answers
53 views

Prove the identity $\sum_{k=0}^{n}\sum_{r=0}^{k} \binom{k}{r} \binom{n}{k} = 3^n$

Prove the identity $\sum_{k=0}^{n}\sum_{r=0}^{k} \binom{k}{r} \binom{n}{k} = 3^n$. I believe I need to use the binomial theorem here, but I don't know how to deal with the double summations.
-2
votes
2answers
36 views

Find the S value - Logarithmic summation [on hold]

Find the value $S$. $$ S = \frac{1}{\log_2 1000!} + \frac{1}{\log_3 1000!}+\frac{1}{\log_4 1000!}+...+\frac{1}{\log_{999} 1000!} + \frac{1}{\log_{1000} 1000!} $$ Any ideas?
0
votes
2answers
27 views

Why the summation of one expression times $x^n$ times another expression times $x^n$ equals this?

Why $$ \sum ^{\infty }_{n=0}a_{n}x^{n} \sum ^{\infty }_{n=0}b_{n}x^{n} = \sum ^{\infty }_{n=0}\left( \sum ^{n}_{k=0}a_{n-k}b_{k}\right) x^{n} $$
0
votes
0answers
62 views

On the integral $\mathfrak{I}~=~\int_0^{\pi}\cos(x)^{\sin(x)}dx$

While thinking about this recent question I thought about an attempt utilizing derivatives of the Beta Function. Sadly I realized that it does not work out for the linked integral but possible for a ...
-1
votes
0answers
19 views

Substitute sum in another sum

I arrived at this point, having P defined as: $P = \sum_{x \in L} \mu(2x-\mu)$ where $\mu = \dfrac{1}{|L|}\sum_{v\in L}v$ Now how can I replace $\mu$ in P? I would like to have only one sum in the ...
3
votes
2answers
67 views

Find close expression for the sum $S_{n,k}=\sum\limits_{i=0}^{2n} (-1)^i \binom{n-1}{i} \binom{n+1}{k-i}.$

Find close expression for the sum $$S_{n,k}=\sum_{i=0}^{2n} (-1)^i \binom{n-1}{i} \binom{n+1}{k-i}.$$ For small $k$ I have got following \begin{align} &S_{n,0}=1,\\ &S_{n,1}=2,\\ &S_{n,2}=-...
0
votes
0answers
43 views

Sum of quadratic series [duplicate]

I just found out that $$\sum _{i=0}^n\:i^2= \frac{n\left(n+1\right)\left(2n+1\right)}{6}$$ and $$\sum _{i=0}^n\:i^3 = \frac{n^2\left(n+1\right)^2}{4}$$ but I don't understand how did we get the ...
-4
votes
1answer
29 views

Riemann rearrangement theorem and Leibniz summation [on hold]

Let $s = \sum \limits_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ Show that $1+\frac{1}{3}-\frac{1}{2} +\frac{1}{5}+\frac{1}{7}-\frac{1}{4} +\cdots = \frac{3s}{2}$ ?
0
votes
0answers
41 views

how can i prove $\sum_{t=1}^T a_t.n_t > \sum_{t=1}^T a_t.n'_t $?

in partitioning of numbers (ways of writing a positive integer as a sum of positive integers) . suppose that$ P $and $Q$ are two partitions of $N \in \mathbb N$. $n_t$:=the number of times that a ...
0
votes
0answers
65 views

Evaluating the continued fraction

How to evaluate $\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\frac{1}{a_{4}+\frac{1}{\ddots}}}}}$, where $a_{j}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{j}$? That is, How to evaluate ...
1
vote
2answers
32 views

Is it true that $x\geq \sum_{k=1}^\infty k \prod_{n=1}^k \frac{x}{x+n}$ for $0<x<1$

I would like to prove or disprove $x\geq \sum_{k=1}^\infty k \prod_{n=1}^k \frac{x}{x+n}$. From numerical examples it looks like the inequality holds. However, I have no idea how to prove it. I have ...
1
vote
1answer
20 views

Explanation of summation equation

Could somebody please explain the following equation to me? I have no clue what H represents, nor how theta(ln(n)) - theta(ln(k)) results in theta(ln (n/k)) Any explanation would be appreciated. ...
0
votes
0answers
34 views

This semi-harmonic-series converges [duplicate]

We know that the series $\sum_{k=1}^{\infty}\frac{1}{k}$ diverges. The series $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{8}+\frac{1}{10}+\frac{1}{11}+\dots+\frac{1}{18}+\frac{1}{20}+\...
-1
votes
1answer
83 views

Evaluation of Euler-type sum $\displaystyle\sum_{n=1}^{\infty}\frac{H_{n}^{2}}{(2n+1)^{2}}$ [on hold]

How can one evaluate the sum $\displaystyle\sum_{n=1}^{\infty}\frac{H_{n}^{2}}{(2n+1)^{2}}$? Here $H_{n}$ denotes $n$-th harmonic number.
1
vote
1answer
17 views

Sigma notation for iterating through number of members of a set with constant expression

Say I have a graph G and I want to sum some constant C (like the minimum degree of the graph) for every vertex. Can I use the following notation? $$\sum_{x \in V(G)}C $$ Is this an appropriate way ...
0
votes
1answer
29 views

Dirichlet test for complex sequences

The sum $$\sum_{k=1}^\infty a_k b_k$$ converges when $a_k$ is monotonically decreasing and $$B_n=\sum_{k=1}^n b_k$$ is finite/bounded $\forall n$. This follows from summation by parts. I'm now ...
0
votes
0answers
30 views

Evaluation of the Euler type sum [duplicate]

How does one evaluate the sum $\displaystyle\sum_{n=1}^{\infty}\frac{H_n}{(2n+1)^{2}}$, where $H_n = \sum_{k=1}^{n} \frac 1k$?
0
votes
3answers
24 views

shifting indices of summation

how come you can rewrite $$\sum\limits_{j=i}^{n} j = \sum\limits_{j=0}^{n-i} (i+j) = \sum\limits_{j=0}^{n-i} i + \sum\limits_{j=0}^{n-i} j$$ I don't understand how the indices were shifted and why you'...
2
votes
1answer
29 views

Fractional part distribution

It is known that the distribution of $\{\sqrt{n} \}$, evaluated over the integer values of $n$, is uniform in the interval $[0,1)$. Let us consider the sum $$S(K)=\sum_{n=1}^K \left(\{\sqrt{n}\}-\...
0
votes
0answers
33 views

Sum of hypergeometric function

I am trying to evaluate the following sum $$\sum_{n=1}^N {}_2F_1(-n,n-N,1,x) y^n $$ I notice that according to wolfram alpha, $$\sum_{n=1}^\infty {}_2F_1(-n,b,c,x) y^n = \frac{_2F_1(1,b,c,\frac{...
21
votes
2answers
2k views

A simple finite combinatorial sum I found, that seems to work, would have good reasons to work, but I can't find in the literature.

I was doing a consistency check for some calculations I'm performing for my master thesis (roughly - about a problem in discrete bayesian model selection) - and it turns out that my choice of priors ...
0
votes
1answer
27 views

Riemann sums over dense countable sets

Let $f$ and $g$ be positive, smooth and integrable functions in $\mathbb{R}$, whose derivatives are also integrable. Assume as well that the expression $$ \frac{\sum_{q\in \mathbb{Q}} f(q)}{\sum_{q\in ...
1
vote
3answers
56 views

value of $k$ in binomial expression

If $\displaystyle \binom{404}{4}-\binom{4}{1}\cdot \binom{303}{4}+\binom{4}{2}\cdot \binom{202}{4}-\binom{4}{3}\cdot \binom{101}{4}=(101)^k.$ Then $k$ is Iam trying to simplify it $\displaystyle \...
-3
votes
0answers
35 views

Summation of infinite integers. [on hold]

I came across this while preparing for national level exam. I couldn't figure out how to think on it. Can anyone help to analyse what each option says and why its right/wrong ? with some basic points ...
-1
votes
1answer
13 views

Simplification of Summation for an Expected Value Problem

I am not good with series or summation notation. Could someone explain how they simplified the expression in this image. Thanks!
0
votes
2answers
32 views

evaluation of this sum?

$$\sum\limits_{j=i}^{n} j = i + (i+1) + (i+2) + \cdots + (i+k)$$ note: above $(i+k) = n$ $$\\$$ now $$\sum\limits_{j=i}^{n} j = i \cdot(k+1) + \frac{(k+1)k}{2} = \frac{(k+1)(2i+k)}{2} $$ so my ...
1
vote
2answers
69 views

How to show $\lim_{n\to\infty}n\left\{\sum_{k=1}^n\frac{1}{(n+k)^2}\right\}=\frac{1}{2}$

Show that $$\lim_{n\to\infty}n\Bigg\{\dfrac{1}{(n+1)^2}+\dfrac{1}{(n+2)^2}+\dfrac{1}{(n+3)^2}+\cdots+\dfrac{1}{(2n)^2}\Bigg\}=\dfrac{1}{2}.$$ Proof: We can rewrite $$\lim_{n\to\infty}n\Bigg\{\...
0
votes
2answers
36 views

How to find $\int_0^1x^3$ using sums and partitions?

The problem statement is to. Calculate $\int_0^1x^3dx$ by partitioning $[0,1]$ into subintervals of equal length. This is my attempt: Let $p=3.$ Let $\delta x = 0.5$ so that the partition is $[0,0.5],...
-2
votes
2answers
30 views

How do I solve this summation? [closed]

$$\sum^{16}_{k = 2} (5-k)$$ How do I solve this summation? I could not try any way because I'm new in this.
0
votes
0answers
10 views

Summation when the stopping point is less than the starting point

I have notation for summation like this, $\sum_{i=n+1}^{m-1} (n+m-i)$, $m>n$. My question is, Does the summation equal $0$ if $m=n+1$ or undefined?? If it is undefined, how should I write ...
0
votes
0answers
13 views

Summation with independent variables as different sized arrays in matlab

I am new to matlab and am learning it on my own. I am trying to solve and graph the following summation in matlab ...
1
vote
2answers
15 views

Summation equivalence

I am having trouble seeing how this summation equivalence holds true: $\sum_{i=0}^\infty x^i$ = $\frac{1}{1-x}$ if |x| < 1 The only thing I can see where there would be a problem is if x = 1 or ...
1
vote
2answers
18 views

Calculating convergence of a sum

I have the sum: $$S=\sum_{n=1}^\infty\frac{n!}{n^n}$$ and I am using D'Alembert's test for convergence which states for some sum: $$\sum_{n=a}^\infty u_n\,\,(a\neq\pm\infty)$$ that it is convergent if:...
0
votes
1answer
40 views

Chapter V: Titchmarsh's book “The theory of the Riemann Zeta function”

Through chapter V of Titchmarsh's book "The theory of the Riemann Zeta function" it is used a "counting technique" that I am not understanding. In particular, Theorem 5.12, p 106, uses something like:...
2
votes
1answer
30 views

Sum of terms with recurrence relation

I have the following sequence, where $s$ is some positive multiple of 4: \begin{equation} L_n = \begin{cases} \frac{(s-2)!}{2^{s/4-1}(s/2)!(s/4-1)!}, & \text{for $n=1$} \\ \\ L_{n-1}\cdot\frac{2(...
-1
votes
1answer
13 views

what is the order of growth of the following sum? [closed]

what is the order of growth of the following sum ? $$ \sum_{i=1}^n \left(i^2+1\right)^2 $$
0
votes
0answers
26 views

Intuitive method for solving $\sum_{k=1}^{n} k! * \sum_{k=1}^{n} k$

Is there any intuitive method of solving: $\sum_{k=1}^{n} k! * \sum_{k=1}^{n} k$ without having to develop the Exponential Integral Ei?
0
votes
1answer
34 views

How to calculate this $\sum\limits_{n=0}^{\infty}\frac{n}{2^n}e^{jwn}$

How to calculate this $\sum\limits_{n=0}^{\infty}\frac{n}{2^n}e^{-jwn}$,because it is not geometric progression,so i can't know how to solve it,can anyone help me?
0
votes
0answers
26 views

Are There Any Special Properties for Different Number of “Terms” in Equal Summations

I found that solving for $e^{ix}$ gave a formula that produces two n "terms" of the starting summation and defining terms as two n products like 1+ix being two terms of the summation $e^{ix} $ =$\...
4
votes
2answers
62 views

Asymptotic of sum $\sum_{j=1}^n j^{f(n)}$

What is known about the asymptotic of $\sum_{j=1}^n j^{f(n)}$ where the exponent is some function that grows with $n$? For instance, if $f(n) = k$ is constant, then we know it's $\frac{1}{k+1}n^{k+1} ...
0
votes
2answers
31 views

Trying to prove an equation

I would like to receive some help about the next problem. The problem: I'm trying to prove the next equation: $$\sum_{k = 0}^{n} \frac{(-1)^{-k}}{k!(n - k)!} = 0 \quad, n = 1, 2, ...$$ My work ...
2
votes
2answers
43 views

Prove $\sum_{i=1}^n a_i$ = $\sum_{i=2}^{n+1} a_{i-1}$

Given $\sum_{i=1}^n a_i$ = $\sum_{i=2}^{n+1} a_{i-1}$ How would you show this true for all n ∈ N and $a_1, a_2, . . . , a_n$ ∈ R? I know it is obviously true because i would just use a substitution ...
-1
votes
1answer
38 views

Sum of products of odds over evens. [closed]

I have been trying to solve this summation but it eludes me: $S = 1 + \dfrac34 + \dfrac{(3\cdot 5)}{(4\cdot6)} + \dfrac{(3\cdot5\cdot7)}{(4\cdot6\cdot8)} + \cdots$ for $n$ terms. Can anyone help?
0
votes
1answer
30 views

Splitting a double summation

I'm trying to figure out double summations. I was wondering, when trying to simplify them. Can I just follow the same rules as with a standard summation? I know that I'm allowed to split a summation, ...
0
votes
0answers
23 views

How do you evaluate an infinite sum to a given decimal point?

Given some infinite sum $S=\sum_n^\infty f(n)$, where $f$ is some algebraic function and $S$ is convergent, are there well defined method(s) for evaluating $S$ to a given decimal point?
8
votes
1answer
179 views

Calculating $\sum\limits_{n=1}^\infty\frac{{1}}{n+3^n} $

I was able to prove this sum $$\sum_{n=1}^\infty\frac{{1}}{n+3^n}$$ is convergent through the comparison test but I don't get how to find its sum.