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

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0
votes
2answers
31 views

Integral of $\sin(x)$ using power series.

$\displaystyle \int_{0}^{1} \sin(x) \, dx$ $\sin(x) = \displaystyle \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$ Integrating this from $0 \to 1$ On the RHS we get $\displaystyle ...
0
votes
1answer
50 views

How to solve integrals using series?

Many places I have seen when solving integrals you change a lot of it into sums. Finding $\int_{0}^{\pi/2} \dfrac{\tan x}{1+m^2\tan^2{x}} \mathrm{d}x$ Is just an example. So in general, how do you ...
1
vote
0answers
40 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
49 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
41 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 ...
2
votes
1answer
61 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
196 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
28 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
256 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
22 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
48 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
21 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 ...
6
votes
1answer
69 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 ...
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
1answer
261 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?
11
votes
3answers
182 views

Proving this formula $1+\sum_{n=0}^{\infty }\frac{1}{\pi \left(2n+\frac{3}{4}\right)\left(2n+\frac{5}{4}\right)}=\sqrt2$

I tried to prove this formula but I couldn't do. $$1+\sum_{n=0}^{\infty }\frac{1}{\pi \left(2n+\frac{3}{4}\right)\left(2n+\frac{5}{4}\right)}=\sqrt{2}$$
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
70 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
36 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
66 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
29 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
19 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
69 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
14 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
140 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
148 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
53 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
66 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
121 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
54 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
42 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
34 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
33 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
92 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
21 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
116 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
70 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
61 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
118 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
48 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 ...