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

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1
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2answers
46 views

How to evaluate $ (0.3)^n\sum_{m=0}^n \left(\frac{0.8}{0.3}\right)^m $?

Can someone please help me solve this sum: $$ (0.3)^n\sum_{m=0}^n \left(\frac{0.8}{0.3}\right)^m u[n] $$ where $u[n]$ means just that $n \ge 0$. I keep getting $$-2(0.3^n -(0.8^(n+1))/(0.3))$$ but ...
1
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1answer
91 views

Proof that $\sum\limits_{i=1}^n \cos \sqrt{i}$ is unbounded. [on hold]

Please advice how to prove that $\sum\limits_{i=1}^n \cos \sqrt{i}$ is unbounded. By this I mean there exists no positive real $B$ such that for any natural $n$ $$-B <\sum\limits_{i=1}^n \cos ...
0
votes
0answers
38 views

Help in simplifying this bad looking expression

Can I express the following $$\sum_{(i,j)\in\mathcal{R}} A_{v_i} G(v_j-v_i)$$ where $\mathcal{R}=\{ (i,j) \in \mathbb{Z}^2,1\leq i\leq n, 1\leq j\leq m\}$ while $G(.)$ is any function of (.) and ...
0
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1answer
28 views

Arithemetic series addition

Lets say I have M= 1+2+3+4+5+6+7.... (to infinity) and I have another sequence,N= 6+14+22+30..... (to infinity) is it possible to say that N = 4M +2 ? Or is there another way that I can write ...
20
votes
1answer
233 views
+100

Using Fourier Series to compute sums

I have just started learning the basics of Fourier series and have some doubts about it. I am aware that Fourier series can be used to compute infinite sums. For example, $\zeta(2)$ and $\eta(2)$ can ...
4
votes
2answers
96 views
+50

Closed form of $\sum_{k=0}^{\infty} \frac{k^a\,b^k}{k!}$

While working on this question I think I've found a closed-form expression for the following series, but I don't know how to prove it. Let $a \in \mathbb{N}$ and $b \in \mathbb{R}$. Then ...
1
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1answer
28 views

Proof that $ y(n) = ∑_{k=-∞}^{∞}\ {a}^{-k}u(n-k)u(-k) = \frac{1}{1-a }$ if $n>0$

Can someone explain the steps and how the boundaries for the summation change to result in the answer (And possible for the case where $n\leq 0$. I am not really a mathematician, don't know if the ...
2
votes
6answers
40 views

Notation for sum of products

Is there a summation notation for the sum of products made two by two? I have the following expression: $$x_1x_2+x_1x_3+\dots+x_1x_n+x_2x_3+x_2x_4+x_2x_5+\dots+x_2x_n+\dots+x_{n-1}x_n$$
5
votes
2answers
98 views

Evaluating a series of hypergeometric functions

I would like to prove (or disprove) the following statement: $$ \sum_{n=0}^\infty \left[\frac{{}_2{\rm F}_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};1\right)}{n!}\right] = \frac{\pi}{2} \left[ ...
0
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0answers
19 views

Determine f(z) by evaluating the sum

Determine an explicit expression for $f(z)$ by determining the sum of the series $f(z) = \sum_{n = 1}^\infty \frac{1}{n}$ $\cdot (\frac{z}{z-1})^n$ where $z\ne 1$ Yeah... I really don't know where ...
0
votes
2answers
19 views

How to find sum of powers from 1 to r

Let say I have two numbers n power r. How can we find sums of all powers. For example if n = 3 and r 3 then we can calculate manually like this ...
2
votes
2answers
59 views

Factorial as a sum. Insight appreciated

I recently posted an answer to a question about ways to express the factorial function as a sum. I posted the following formula, which I discovered several years ago and I haven't seen anywhere else: ...
1
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1answer
37 views

Evaluating $ \sum_{i=0}^{\infty}ir^i$ for $|r|\lt 1$ [duplicate]

I'm doing my mathematics homework and there's question which I'm pretty much unable to solve. I've literally tried every method but no results. It'd be great, if anybody could help me. Thanks (in ...
6
votes
6answers
218 views

Prove that $\sum_{n=1}^\infty \frac{n^2(n-1)}{2^n} = 20$

This sum $\displaystyle \sum_{n=1}^\infty \frac{n^2(n-1)}{2^n} $showed up as I was computing the expected value of a random variable. My calculator tells me that $\,\,\displaystyle ...
1
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0answers
25 views

Len $a_n$ be infinite sequence such that $|{a_{n+1}}|<|a_n|$. Assumbe that $S_{2^n}$ is bounded. Does it imply that $S_n$ is bounded too?

Len $a_n$ be infinite sequence such that $|{a_{n+1}}|<|a_n|$ and let $S_n = \sum\limits_{i=1}^n{a_i}$. Assumbe that $S_{2^n}$ is bounded, i.e. there exists positive $B$ such that for any natural ...
5
votes
3answers
125 views

Proof of Nesbitt's Inequality?

I just thought of this proof but I can't seem to get it to work. Let $a,b,c>0$, prove that $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge \frac{3}{2}$$ Proof: Since the inequality is homogeneous, ...
0
votes
1answer
12 views

Simplification of a large sum obtained from the 1-D wave equation

I have acquired the sum below through Fourier, and was wondering if there was anyway to simplify it, since it is large and ugly. $$\sum \limits_{n=1}^\infty \frac{-2K_1}{n\pi} ...
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0answers
12 views

How do I evaluate a summation for its comlexity? [on hold]

Here, question #1 the answer is O(n^3), how do you solve for that? Why isn't it O(n)?
0
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7answers
161 views

$\displaystyle\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$

Show that $\,\displaystyle\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$. I'm thinking right now (though not getting anywhere with it) that I want to expand out the summation portion to ...
-1
votes
2answers
35 views

Geometric summation proof, not calculus

I am trying to take the expression $$T=\sum_{k=1}^nkx^k$$ and make it into a "simpler expression." I have an example similar to it where i am finding $$\sum_{k=1}^nx^k$$ where the answer is $$S_0 = ...
15
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4answers
437 views
+50

Calculate $\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$

I'm an eight-grader and I need help to answer this math problem. Problem: Calculate $$\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$$ This one is very hard for ...
1
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3answers
21 views

summation and product of sin and cos

I wonder how to find summation for $\displaystyle \sum_{k=0}^{n-1}(\cos{\frac{2\pi k}{n}+i \sin\frac{2\pi k}{n}})$ and the same for product $\displaystyle \prod_{k=0}^{n-1}(cos{\frac{2\pi k}{n}+i ...
0
votes
3answers
33 views

Limit of a sum as n approaches infinity.

I have a sum $$T_n(x)= 1 + 2x + 3x^2 + \dots +n x^{n−1}.$$ and I am supposed to find the limit of $T_n(1/44)$ as $n$ approaches infinity. I would appreciate any suggestions on how to proceed.
2
votes
2answers
54 views

Proof of a summation of $k^4$

I am trying to prove $$\sum_{k=1}^n k^4$$ I am supposed to use the method where $$(n+1)^5 = \sum_{k=1}^n(k+1)^5 - \sum_{k=1}^nk^5$$ So I have done that and and after reindexing and a little algebra, ...
2
votes
2answers
49 views

How to make a sum vanish?

This is a very very basic question but I just cannot think of a way to tackle it for some reason. Say I have three numbers $a,b,c$ with the sum $a+b+c\neq1$. Now if I want to make this sum equal to 1 ...
0
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2answers
16 views

finding the largest term in a binary summation

I'm working on a problem that involves the following summation: $$y=\sum_{i=0}^{x}i2^i$$ I need to determine the largest value of $x$ such that $y$ is less than or equal to some integer K. Currently ...
0
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2answers
37 views

Solving an unusual equation

I need to find a real number $n$ such that $n > 1$ and: $$ \sum_{k=1}^\infty \frac{2^k}{n^k} = \frac{n-1}{n} $$ Ideally, I'd find the minimum such $n$ (if more than one exists), but really, any ...
2
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0answers
41 views

Convergence of $\sum_{n=1}^{\infty} \frac{Sin^2(n)}{n}$ [duplicate]

Is following sum convergent? $$\sum_{n=1}^{\infty} \frac{Sin^2(n)}{n}$$ Integral test, Dirichlet test doesn't apply. Any idea !
0
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0answers
32 views

a complicated Sum

I don't know how i can calculate this complicated multivariate sum : $$ S(l,k)=\sum_{|m|=k} s_{l,m}=\sum_{|m|=k} l(l-1)(l-2)\dots(l-m+1) $$ Where $m=(m_1,\dots,m_n)$, $l=(l_1,\dots,l_n)$, and $k$ a ...
1
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1answer
89 views

Is the series: $\,\sum_{n=1}^{\infty}\frac{\mathrm{e}^{-in}}{n}\,$ divergent?

According to mathematica, the complex series $\displaystyle\sum_{n=1}^{\infty}\frac{e^{-in}}{n}$ does not converge. I know that the factor $\dfrac{1}{n}$ in the above series is diverging, but I don't ...
1
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5answers
39 views

Intuition to why average of the square of a positive integer and the integer itself is the sum of all numbers from 1 to the integer?

The sum of all numbers from 1 to n, i.e. $\sum_{i=1}^n i = \frac{n(n+1)}{2} = \frac{n^2 + n}{2}$ This happens to be show that the average of a number and its square equals the sum of all numbers ...
1
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2answers
42 views

Calculus add formula to derive new formula

I was asked to re-write a formula forward and backward and derive a new formula from it. Here's the problem: Write $$S=1+2+3+\cdots+N$$ forward and backward $$\begin{array}{rcrcrcr} S & = ...
2
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2answers
65 views

Question on simplification of $\sum_{n=1}^{\infty}\frac{2}{(2n+1)(2n+3)}$?

I am having trouble seeing how $\sum_{n=1}^{\infty}\frac{2}{(2n+1)(2n+3)}$ equals $\sum_{n=1}^{\infty}\frac{1}{2n+1}-\frac{1}{2n+3}$. I can see $\sum_{n=1}^{\infty}\frac{1}{2n+1}+\frac{1}{2n+3}$ but ...
2
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1answer
33 views

Nice derivation of $\sum_{n=1}^\infty \frac{1}{n} \left( \frac{q^{2n}}{1-q^n}+\frac{\bar q^{2n}}{1-\bar q^n}\right)=-\sum_{m=2}^\infty \ln |1-q^m|^2$

I'm searching for a nice derivation of the formua $\sum_{n=1}^\infty \frac{1}{n} \left( \frac{q^{2n}}{1-q^n}+\frac{\bar q^{2n}}{1-\bar q^n}\right)=-\sum_{m=2}^\infty \ln |1-q^m|^2$ given for example ...
0
votes
1answer
13 views

Binary representation of the real numbers

I am solving the following exercise: for $n \in \mathbb{N}$ and $a_1,a_2, \ldots ,a_n \in \{0,1\}$ we define: $$ I(a_n, \ldots , a_n) := \left \lbrack \sum_{i=1}^n \frac{a_i}{2^i}, ...
2
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1answer
110 views

Combination of quadratic and cubic series

I'm an eight-grader and I need help to answer this math problem (homework). Problem: Calculate $$\frac{1^2+2^2+3^2+4^2+...+1000^2}{1^3+2^3+3^3+4^3+...+1000^3}$$ Attempt: I know how to ...
7
votes
1answer
105 views

The meaning of a definition involving multiple sums with Bernoulli numbers

Reading a paper regarding Bernoulli numbers, and I stumbled onto a definition. First let $$\frac{x}{e^x-1}=\sum_{k=0}^{\infty}B_k\frac{x^k}{k!}$$ The author then goes on to define new terms. Let ...
0
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1answer
11k views

Solution of Introduction to Algorithms 3rd edition (Cormen)

I'm studying algorithms by Cormen book. I have already find a pdf with the answers of the questions. www2.compute.dtu.dk/~phbi/files/teaching/solution.pdf But it isn't have the all solutions. I'm ...
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votes
2answers
28 views

Derive a Formula for the following sum. [closed]

Let abs(r)<1 be a real number. Evaluate sum(i*(r^i)) from i=0 to infinity.
2
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4answers
45 views

Closed Form for Factorial Sum

I came across this question in some extracurricular problem sets my professor gave me: what is the closed form notation for the following sum: $$S_n = 1\cdot1!+2\cdot2!+ ...+n \cdot n!$$ I tried ...
1
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3answers
37 views

Summation Proof

I'm getting stuck halfway through this: Show that $$\sum_{i=1}^n (y_i - \bar y_s)^2 = \sum_{i=1}^n (y_i)^2 - n\bar y_s^2$$ My skills with manipulating sums are quite rusty. I multiply the left side ...
2
votes
2answers
19k views

How to get to the formula for the sum of squares of first n numbers? [duplicate]

Possible Duplicate: How do I come up with a function to count a pyramid of apples? Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$? Finite Sum of Power? I know that the sum ...
2
votes
4answers
96 views

Sum of the series $\frac{(-3)^{n-1}}{8^n}$

It might looks obvious to you but I don't manage to find the sum: $$\sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{8^n}$$ Can anyone help me ? Thanks
1
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3answers
30 views

Compute the following sum for any x?

Compute the following sum for any x? $\sum_{n=0}^\infty {(x-1)^n\over (n+2)!}$ I am having trouble to compute that sum. It looks like geometric series but I don't know where to start. Can everyone ...
8
votes
2answers
78 views

Exactly expressing integral as a sum

Apparently (i.e. according to my professor), the following holds:$$\int_a^b f(x) dx = (b-a)\sum_{n=1}^\infty \sum_{m=1}^{2^n-1} (-1)^{m+1}2^{-n}f(a+m(b-a)2^{-n}).$$How would one go about proving such ...
3
votes
2answers
86 views

A double Summation involving 7th roots of unity

Is there possibly a closed form for $$\sum_{m=1}^{\infty} \left(\sum_{k=1}^{6} \dfrac{1}{m-\alpha^k}\right)^2$$ where $\alpha=e^{2\pi i/7}$ ? I'm having problems evaluating the first sum, let alone ...
0
votes
0answers
20 views

Expression for a series of squared sines

Does anyone know if there is a single expression for $$-\frac{1}{2}\sum_{j=1}^{\infty}\sin^2\left(\frac{2\pi x}{3^j}\right)$$ or at least a nicely-expressed upper bound? I've already computed that ...
1
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1answer
24 views

Find the sum of n terms of a series

Find the sum of n terms of series whose $n$th term is $\frac{n^2(n^2-1)}{4n^2-1}$.
0
votes
0answers
32 views

Minimizing sum of weighted product

Consider a total of $d$ items, $\{I_1,I_2, \cdots,I_d\}$, each having a weight $w_i$ (a positive integer), and a total of $m$ bins, $\{B_1,B_2,⋯,B_m\}$. We would like to distribute the items into the ...
2
votes
5answers
84 views

Prove that $\left(\sum_{k=1}^{n}k\right)^2=\sum_{k=1}^{n}k^3$ holds true for $n ≥ 1$

I've been trying to figure out this proof for way too long now, I'm just not sure where to begin for the inductive step. Any guidance would be greatly appreciated.