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

learn more… | top users | synonyms

1
vote
0answers
34 views

The sum of finite exponential series with a quadratic phase

How can I prove that: $$ \sqrt \frac K2 + i \sqrt \frac K2=\sum^K_{m=1}\exp\left(i\frac \pi Km^2\right) $$ When $K$ is even.
0
votes
2answers
47 views

What is the simplified of this summation?

How can I simplify this equation into a single equation in terms of $n$? $$\sum_{k=0}^{n-2}{(n-k-1)(n-k-2)+k(k+1)}$$
1
vote
0answers
19 views

How to to minimize a sum by changing summation order

I have two vectors $(x_1,\dots,x_n),(y_1,\dots,y_n) \in \mathbb{R}^{n}$. I want to find a permutation $\sigma$ such that $$ \sum_{i=1}^n |x_i -y_{\sigma(i)}|^2$$ is minimized. Is there a better way ...
3
votes
0answers
38 views

Nice formula for $\sum_{m=0}^n{2m\choose m}{2(n-m)\choose n-m}$? [duplicate]

I am trying to find a nice formula for \begin{align}\sum_{m=0}^n{2m\choose m}{2(n-m)\choose n-m}\tag{1}.\end{align} After failing to simplify it, I asked WolframAlpha (see link), and apparently, it ...
1
vote
0answers
34 views

Prove that the sum of harmonic series 1..n can be expressed as (n+1)H_n -n

Prove by induction that the sum of harmonic series Hn from 1 to n where n is a natural number is as follows. $$ H_n = \sum\limits_{i=1}^n 1/i $$ Prove: $$ \sum\limits_{i=1}^nH_i = (n+1)H_n -n $$ ...
2
votes
4answers
190 views

Unable to find the sum of a series

I am trying to find the sum of the following series: $$\sum_{n=1}^{\infty} {\frac{1+7^n}{9^n}}$$ which I rewrote as $$\sum_{n=1}^{\infty} \left(\frac{1}{9^n}+ \left(\frac{7}{9}\right)^n\right)$$ ...
1
vote
1answer
25 views

Convergence of a summation

How do I find out what this summation converges to? I dont even know how I'd start o_o I assume the first part converges to infinity but dont know how the cos works in this case. ...
10
votes
2answers
250 views

$\sum x_{k}=1$ then, what is the maximal value of $\sum x_{k}^{2}\sum kx_{k} $

Let $1\geq x_{1}\geq x_2\geq\cdots\geq x_{n}\geq0$, and $\sum\limits_{k=1}^{n}x_{k}=1$. then what is the maximal value of ? $$\sum_{k=1}^{n}x_{k}^{2}\sum_{k=1}^{n}kx_{k} .$$ I think, Maybe we could ...
0
votes
1answer
21 views

Deriving a formula to find the sum of a series.

I have attempted to solve this problem: Find the sum of the series, if it converges. $\sum\limits_{n=1}^{\infty}\frac{(-2)^{n-1}}{7^n}$ I see that the values of $a_n$ are $\frac{1}{7} + ...
6
votes
1answer
45 views

Finding a bound for $\sum_{n=k}^l \frac{z^n}{n}$

For $z\in\mathbb{C}$ such that $|z|=1$ but $z\neq1$ and $0<k<l$, I'm trying to prove that: $$\left|\sum_{n=k}^l \frac{z^n}{n}\right| \leq \frac{4}{k|1-z|}$$ It's more of a game that slowly ...
0
votes
1answer
47 views

Prove that $\lim_{n\to \infty} \sum_{i=1} ^n\frac{1}{n+i}=\ln2 $ [closed]

Need help with the following proof. $$\lim_{n\to \infty} \sum_{i=1} ^n\frac{1}{n+i}=\ln2 $$ Its the night before my maths exam. I know its a silly question but I need to prove whether given equality ...
0
votes
1answer
20 views

Sum notation of a tuple or set

I'm currently confused in how to express the sum of a tuple. I have a set or a tuple (for summation, order shouldn't be an issue) like this: $$S_{A,B,C} = (1,3,6)$$ The subscript $A,B,C$ has nothing ...
-2
votes
0answers
10 views

Sum of k/GCD(k,n)

I want to find sum of $\sum_{1}^{n}\frac{k}{gcd(k, N)}$ Here $k$ varies from $1$ to $N$ and $N$ is from $1≤ N ≤ 10^{10}$ . I don't know how to do this please help? Thanks in Advance.
6
votes
1answer
67 views

A limit related to the $\zeta(3)$ and the fractional part

I need some clues, hints for proving that $$\lim_{n\to\infty} n\frac{\displaystyle \left\{\frac{n}{\sqrt{1}}\right\}+ \left\{\frac{n}{\sqrt{2}}\right\}+ \left\{\frac{n}{\sqrt{3}}\right\}+\cdots ...
23
votes
4answers
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
1answer
255 views

Calculate the infinite sum $\sum_{k=1}^\infty \frac{1}{k(k+1)(k+2)…(k+p)} $

I have to prove that $$\sum_{k=1}^\infty \frac{1}{k(k+1)(k+2)....(k+p)} $$ is equal to $\, \dfrac{1}{(\,p-1)!p}.$ How can I do that?
1
vote
1answer
47 views

Prove Jensen's inequality

$$\left(\sum_{i=1}^na_i^p\right)^{1/p} \ge \left(\sum_{i=1}^na_i^q\right)^{1/q} $$ if $0 < p \le q$ for $a_i\ge 0$. I have proved that the inequality holds for $ p=q $ (trivial) and I have also ...
1
vote
3answers
67 views

Proving $\sum^{n}_{k=1} \frac{1}{\sqrt{k}}>\sqrt{n}$ by induction

Prove that $$\sum^{n}_{k=1} \frac{1}{\sqrt{k}}>\sqrt{n}$$ for all $n\in \mathbb{N}$ where $n\geq2$. I've already proven the base case for $n=2$, but I don't know how to make the next step. Is the ...
2
votes
2answers
25 views

Evaluating sum of $\sum_{i=0}^{n} 2^{i/2}$

$$\sum_{i=0}^{n} 2^{i/2} = (1+ \sqrt2)\left(2^{\frac{n+1}2} -1\right)$$ I know the above is true, but how would I get the right hand side? This summation shows up from a algorithm recurrence problem ...
2
votes
1answer
35 views

Integral approach for infinite sum of $e^{-n}$

A while ago a posted this same problem, I have a different approach, just need a little help... $$\displaystyle e^{-n} = \int_{0}^{e} -nx^{-(n+1)} \,dx$$ Originally, we had, $\displaystyle ...
1
vote
2answers
61 views

Solving $T(n)= 2T(n/2) + \sqrt{n}$ without master theorem (algebraically & recurrence tree)

$$T(n)= 2T(n/2) + \sqrt{n}$$ This recurrence was in a stackoverflow question, and I want to solve it without relying on the master method. The solution was given, but wolframAlpha gives a slightly ...
0
votes
2answers
45 views

Prove the inequality.

$$\left(\sum_{i=1}^na_i^p\right)^{1/p} \ge \left(\sum_{i=1}^na_i^q\right)^{1/q} $$ if $0 < p \le q$ for $a_i\ge 0$. I have proved that the inequality holds for $ p=q $ (trivial) and i have also ...
0
votes
1answer
34 views

Series termination

The successive terms in a power series are given by the recurrence relation $$\frac{a_{n+1}}{a_{n}}= \frac{n(n-1)+\lambda}{9(n+1)(n+2)}$$ where $\lambda=\text{const.}\in\mathbb{R}$. So our power ...
1
vote
2answers
28 views

Sum convergence

I want to check this sum: $$\sum\limits_{n=1}^\infty (\frac{(3n+1)!}{n!(2n+1)!}*7^{-n})$$ I think, the easiest way is to use the ratio test: ...
0
votes
1answer
18 views

Summation manipulation

I want to show $$\sum^{\infty}_{k=1} \frac{x^{2k-1}}{2k-1} \equiv \frac{1}{2}\left[\sum^{\infty}_{n=1}\frac{x^{n}}{n} - \sum^{\infty}_{n=1}\frac{(-x)^{n}}{n}\right]$$ Although I'm really not sure ...
-2
votes
4answers
62 views

Sum of a series - e [closed]

i have this series: $\sum\limits_{n=1}^\infty \frac{13^n}{n!} $. I know, that $\sum\limits_{n=1}^\infty \frac{1}{n!} =e-1$. But what is here the sum? Thank you
0
votes
0answers
12 views

Integration matrix

I want to do integration(summation) of a signal(x) using matrix multiplication. I am looking for a transformation matrix, I corresponding to integration such that F = I * x , where x is the signal ...
0
votes
4answers
134 views

A closed form for sum of binomial coefficients

What is a closed form of the sum: $$\binom{n}{0}+\binom{n-1}{1}+\binom{n-2}{2}+\binom{n-3}{3}+\cdots$$ A combinatorial proof would also be much appreciated. Any general techniques to solve such sums ...
8
votes
4answers
143 views

Proof that $\sum_1^{\infty} \frac{1}{n^2} <2$

I know how to prove that $$\sum_1^{\infty} \frac{1}{n^2}<2$$ because $$\sum_1^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}<2$$ But I wanted to prove it using only inequalities. Is there a way to do ...
3
votes
1answer
52 views

Does this simple sum converge

I'm trying to determine whether the sum $$S=\frac{2}{1}+\frac{2\cdot 5}{1\cdot 5}+\frac{2\cdot 5\cdot 8}{1\cdot 5\cdot 9}+...+\frac{2\cdot 5\cdot 8...(3n-1)}{1\cdot 5\cdot 9...(4n-3)}+...$$ converges ...
0
votes
3answers
58 views

Evaluate infinite sum for $\frac{1}{n^4}$ using integration

$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^4}$ I want to evaluate this sum by the use of integration. $\displaystyle \int_{\frac{1}{n^4}}^{\frac{4}{n^4}} 1 \space dx = \frac{4}{n^4} - ...
6
votes
4answers
118 views

Calculate $\sum_{n=1}^{\infty}(\frac{1}{2n}-\frac{1}{n+1}+\frac{1}{2n+4})$

I am trying to calculate the following series: $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)}$$ and I managed to reduce it to this term ...
1
vote
0answers
50 views

Simplify the product of two sums

How can I simplify the following product of two sums: $$ \biggl(\, \sum ^{n}_{k=0}a_{k}\biggr) \biggl(\, \sum ^{n}_{k=0}\dfrac {1}{a_{k}}\biggr) $$
0
votes
1answer
25 views

Computing $\sum\nolimits_{n = 1}^{+\infty} n \cdot 0.3^{n - 1}$ with the help of an integral

So, a friend of mine told me the professor in one of his classes had computed the sum $\displaystyle \sum\limits_{n = 1}^{+\infty} n \cdot 0.3^{n - 1}$ by replacing (?) the summation with an integral. ...
2
votes
1answer
41 views

Probability of $B$ winning a series of games

$A$ and $B$ are two players. The probability of $A$ winning a particular game against $B$ is $1/3$ and the probability of $B$ winning the game is $2/3$. They play a series in which the rules are ...
5
votes
1answer
32 views

Solving 2nd order ODE with Frobenius method - problems with summation symbol

I'm trying to solve the ODE: $$ y''(x) + \frac{2x}{(x-1)(2x-1)} y'(x) - \frac{2}{(x-1)(2x-1)} y(x) = 0 $$ I'm trying to find a solution by the Frobenius method, expanding a power series of the ...
0
votes
1answer
30 views

Sum of trigonometric series $\sum_{m=1}^{N-1} \frac{\sin(4\pi mk/N)}{\sin ^2 (\pi m/N) }$

Anybody has some ideas to prove the following identity? \begin{equation} \sum_{m=1}^{N-1} \frac{\sin(4\pi mk/N)}{\sin ^2 (\pi m/N) }= 0 \end{equation} where $N$ is an integer greater than $1$, $k$ ...
1
vote
1answer
91 views

A Simple Bound on Super-Additive Functions

If $f(x)$ is a positive super-additive function ($\sum f(x) \leq f(\sum(x) $), can we prove that: $$I = \sum_i f\left(\sum_j x_{ij}\right) + \sum_j f\left(\sum_i x_{ij}\right) - 2 \sum_i \sum_j ...
4
votes
1answer
100 views

Please calculate $\sum _{ k=0 }^\infty\left[ \tan^{ -1 }\left( \frac { 1 }{ k^{ 2 }+k+1 } \right) -\ldots \right] $

Not many math problems stump me, but this summation has me stumped. Can someone provide a solution to this summation: $$\sum _{ k=0 }^{ \infty }{ \left[ \tan ^{ -1 }{ \left( \frac { 1 }{ k^{ 2 }+k+1 ...
9
votes
1answer
451 views

A question from the dreams realm

Let $\phi:\mathbb{R}\rightarrow\mathbb{R}$ be a function (not necessarily continuous). Let $\phi_0(x)=\phi(x)$ and $\forall k\in\mathbb{N},\phi_{k+1}(x)=\phi(x\cdot\phi_k(x))$. 1. Let ...
0
votes
1answer
18 views

sum of two equal digit numbers vs. sum of those digits

if I take 5688+6984=12672 then sum the result 1+2+6+7+2=18 then sum that result 1+8=9. vs. this. same digits from above. 5+6+8+8+6+9+8+4=54 then sum that result 5+4=9. using this method where the ...
1
vote
2answers
19 views

Intersection of two lines and the minimum of the sum of the two.

We use a formula in my Operations Research class for finding the 'Economic Order Quantity', given the cost function (sum of Holding and Ordering costs) $$C = \frac{Q}{2}H+\frac{D}{Q}S$$ where $Q$ is ...
2
votes
2answers
113 views

Finding the infinite sum of $e^{-n}$ using integrals

I am trying to understand this: $\displaystyle \sum_{n=1}^{\infty} e^{-n}$ using integrals, what I have though: $= \displaystyle \lim_{m\to\infty} \sum_{n=1}^{m} e^{-n}$ $= \displaystyle ...
6
votes
2answers
69 views

Find $S=\frac{1}{2}+\frac{3}{2^2}+\frac{5}{2^3}+\frac{7}{2^4}+…+\frac{2n-1}{2^n}+…$

I'm trying to calculate $S$ where $$S=\frac{1}{2}+\frac{3}{2^2}+\frac{5}{2^3}+\frac{7}{2^4}+...+\frac{2n-1}{2^n}+...$$ I know that the answer is $3$, and I also know "the idea" of how to get to the ...
2
votes
2answers
22 views

Equivalence of summations

Show that $$\frac{1}{n}\sum^{n}_{i=1} (x_{i} - \bar{x})^{2}\equiv \frac{1}{n}\sum^{n}_{i=1}x_{i}^{2} - \bar{x}^{2}.$$ Note that $\bar{x} = \frac{1}{n}\sum^{n}_{i=1} x_{i}$. So I have started by: ...
0
votes
2answers
55 views

find the sum of series

I have problem with finding sum of series: 1)$\displaystyle\sum_{n=k}^{2k-1}\frac{n}{2^n}=?$ 2) $\displaystyle\sum_{n=0}^{k-1}n(\frac{4}{3})^n=?$ I have some idea to 1) ot write it as ...
7
votes
2answers
115 views

Integration of $x^a$ and Summation of first $n$ $a$th powers

There are some convenient formulas for the summation of the first $n$ integers which are the $a$th powers of other integers, e.g. $$ \sum_{i=0}^n i = \frac {n(n+1)}2$$ $$ \sum_{i=0}^n i^2 = \frac ...
5
votes
1answer
47 views

How to prove $\sum_{k=1}^{N} \frac{\sin n\theta}{2^N}=\frac{2^{N+1}\sin \theta + \sin N\theta -2\sin(N+1)\theta}{2^N(5-4\cos \theta)}$

Prove This using De Moivre Theorem $$\sum_{n=1}^{N}\frac{\sin n\theta}{2^n}=\frac{2^{N+1}\sin\theta+\sin N\theta-2\sin(N+1)\theta}{2^N(5-4\cos\theta)}$$ Please help me find my mistake, because ...
4
votes
2answers
19 views

Control ratio of geometric series through its sum

A geometric series $S_n$ is the sum of the $n$ first elements of a geometric sequence $u_n$: $$u_n = ar^n \space \forall n \in \mathbb{N}^*$$ with $u_0$ defined, and: $$S_n = \sum_{k = 0}^{k = n - ...
1
vote
1answer
38 views

Plotting discrete time signals involving sumations in matlab.

Many of the examples I've encountered while looking for an answer are simple functions that do not involve summations. Suppose I have the following function; ...