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

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1answer
12 views

Calculate total revenue share

Suppose you have data such as this: January Total minutes of all videos watched: 50 Total minutes of video X watched: 25 Total revenue: 200 February Total minutes of all videos watched: 200 Total ...
7
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4answers
74 views

Finding the sum of $\sin(0^\circ) + \sin(1^\circ) + \sin(2^\circ) + \cdots +\sin(180^\circ)$

I need help understanding the sum of $\sin(0^\circ) + \sin(1^\circ) + \sin(2^\circ) + \cdots +\sin(180^\circ)$ or $\displaystyle \sum_{i=0}^{180} \sin(i)$ This might be related to a formula to find ...
0
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1answer
18 views

Find $\sum_{i=0}^{\log n} \frac{1}{2^i}$

I'm not really sure how to solve summations, so any help would be great. In particular, I had thought that $n^2\sum_{i=0}^{\log n} \frac{1}{2^i}=O(n^2\log n)$ but it's actually $O(n^2)$, and I'm ...
-1
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1answer
22 views

Set of a summation

Let $S = \{n ∈ N | n \text{ divides the sum of any n consecutive numbers} \}$. How can I describe the set S? I was given the hint: $\displaystyle\sum\limits_{n=1}^N n=\frac{N(N+1)}{2}$ I don't want ...
0
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0answers
6 views

Summation of all combination

I have two matrix. A=[1 2 3];B=[4 5 6]; the all possible combination of their summation is [1+4 1+5 1+6; 2+4 2+5 2+6;3+4 3+5 3+6]. Now instead of 1*3 my matrix dimension is 1*n. and instead of two I ...
2
votes
1answer
64 views

Compute $\sum_{k=0}^{n}\frac{1}{\binom{n}{k}}$

I want to calculate $\sum_{k=0}^{n}\frac{1}{\binom{n}{k}}$. No idea in my mind. Any help? Context I want to calculate the expected value of bits per symbols in adaptive arithmetic coding when the ...
0
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1answer
25 views

summation of series by telescoping series method (feedback needed)

i am stuck i did the first part by cancelling out terms since its a telescoping series. But I do not know how I can proceed any further . Please help. I am not sure of whatever i have done so far. so ...
2
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3answers
121 views

Find if this series converges and if so find its value

I need help I cant understand how we can solve this. I am confused when the log came in. I listed the first few terms but i do not know how to proceed further. all I know is that the sequence is ...
1
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2answers
49 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
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
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6answers
48 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: $$x_1x_2+x_1x_3+\dots+x_1x_n+x_2x_3+x_2x_4+x_2x_5+\dots+x_2x_n+\dots+x_{n-1}x_n$$
0
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0answers
61 views

Help in simplifying this double summation

Can I express the following double summation $$\sum_{(i,j)\in\mathcal{R}} A_{v_i} G(v_j-v_i)$$ where $\mathcal{R}=\{ (i,j) \in \mathbb{Z}^2,i \in [1:n], j \in [1:m]\}$ while $G(.)$ is any function ...
0
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0answers
20 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
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2answers
20 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 ...
0
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1answer
29 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 ...
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 ...
1
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1answer
93 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 ...
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0answers
14 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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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} ...
5
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2answers
101 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[ ...
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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 = ...
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
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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
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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, ...
0
votes
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 ...
1
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1answer
38 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 ...
5
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3answers
126 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, ...
2
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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
38 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 !
4
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2answers
102 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 ...
0
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0answers
33 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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5answers
40 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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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 ...
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 ...
0
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1answer
14 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}, ...
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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
47 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 ...
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 ...
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 ...
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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 ...
3
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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 ...
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
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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
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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.
0
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0answers
25 views

writing sum as a product and vice versa.

$\Pi = k$ from k = 1 to n Can you write this in form of sigma? So that you can evaluate it as a sum? Also, are there any shorthand formula to evaluate a product like there are for summations? ...
0
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1answer
40 views

Prove by Induction : $\sum n^3=(\sum n)^2$ [duplicate]

I am trying to prove that for any integer where $n \ge 1$, this is true: $$ (1 + 2 + 3 + \cdots + (n-1) + n)^2 = 1^3 + 2^3 + 3^3 + \cdots + (n-1)^3 + n^3$$ I've done the base case and I am having ...
0
votes
1answer
24 views

MacLaurin of the Third-degree in sin(a*x)*cos(b*x) at given values

Alright so from my understanding MacLaurin is a special case of Taylor Series but at f(0). However my question involves solving the third degree of MacLaurin for $$f(x) = sin(a \times x)\times ...
0
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0answers
58 views

Evaluating Telescopic Sum $ \sum\frac{n}{1+n^2+n^4} $

$$ \sum_{n=1}^{\infty}\frac{n}{1+n^2+n^4}$$ $x$th Term of this sum can be written as $$T_x=\frac{x}{1+x^2+x^4}=\left(\frac {1}{2\cdot(x^2-x+1)}\right)-\left(\frac{1}{2\cdot(x^2+x+1)}\right)$$ ...