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

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$\phi$, and the uses of an alternate formula

I was trying to find the solution to the formula: $$x = \sum_{n=1}^\infty{x^{-n}}$$ I found it to be the golden ratio, or $\phi = \frac{1 + \sqrt{5}}{2}$. I do not know if this has already been ...
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1answer
28 views

Gauss Method to show [on hold]

Could you please give me the way to solve this problem Using Gauss method to show if $x ≠ y + 1$ then $$ \sum_{i=0}^n (x-y)^i = \frac{(x-y)^{n+1}-1}{x-y-1}. $$
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1answer
26 views

Determine the radius of convergence of $\sum_{n=1}^\infty n^{n^{1/3}}z^n$ (by the ratio test if possible)

Determine the radius of convergence of the following power series: $\sum_{n=1}^\infty n^{n^{1/3}}z^n$ Applying the ratio test gives $\frac{({n+1})^{({n+1})^{1/3}}}{n^{n^{1/3}}}z<1$. So ...
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0answers
12 views

Prove by induction: $E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$ Please just check what I've done

Prove by induction: $$E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$$ Let me show you what I've done. I think I'm right: $$n=1,$$ $$E[c_1U_1(X)] = c_1E[U_1(X)]$$ Okay so maybe this one looks ...
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2answers
55 views

Value of the sum: $\binom{19}{0} - 1/2\binom{19}{1} +1/3\binom{19}{2} - 1/4\binom{19}{3} … -1/20\binom{19}{19}$?

How do I find the value of this sum? I tried taking out the equal binomial coefficients as factors but this didn't really simplify anything. I am stumped.
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2answers
46 views

How we can prove that: $\sum _{_{k=n}}^{^n}\:f\left(\frac{k}{n}\right)\le n\cdot log\left(2\right)$?

$f:\left[0,1\right]\rightarrow R,\:f\left(x\right)=\frac{1}{1+x}$ and we have to show that $\sum _{_{k=n}}^{^n}\:f\left(\frac{k}{n}\right)\le n\cdot\log\left(2\right)$.What I know is just that: ...
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1answer
55 views

How simplify this sum?

I need help to simplify this sum : $$\sum_{i=0}^{x-1}\left(1-\dfrac{1}{2^i}\right)^{m-1}$$ Is it possible to simplify it ? Thank you.
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0answers
46 views

When can we assume this identity about series?

Suppose we are given that $$\sum_{i =a} ^b f (i)g (i) =\sum_{i =a} ^b f (i)h (i) \tag {1} $$ when can we conclude that $$\forall i =a,…,b:g(i)=h(i) \tag {2} $$ ? For example, if $g, h $ are both ...
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2answers
83 views

Find the sum of series $(1^2+1)1!+(2^2+1)2!+(3^2+1)3!+…+(n^2+1)n!$.

Find the sum of series $(1^2+1)1!+(2^2+1)2!+(3^2+1)3!+...+(n^2+1)n!$ I have found one method as i have shown in my answer below. But that form took me 30 mins to identify. ...
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1answer
63 views

evaluate trigonometric

To all the genius out there , here is a question about expresssing summation of hyperboilc functions : First of all, I've already proved that: $$\sinh(x + 1)- \sinh(x) = (-􀀀1 + \cosh (1)) \sinh(x) ...
3
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1answer
53 views

Is there a formula for the closed form for $ \displaystyle \sum_{r=1}^\infty \frac{\sum_{k=1}^r k^n}{r!}$ for any positive integer $n$?

Is there a formula for the closed form for $ \displaystyle \sum_{r=1}^\infty \frac{\sum_{k=1}^r k^n}{r!}$ for any positive integer $n$? I tried Faulhaber's formula and Bell number but couldn't ...
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4answers
128 views

How to prove that the value of $e$ is irrational without using the number $e$ itself [on hold]

Recently I have tried to prove that the value of $e$ is irrational without using the number $e$ itself. I have seen that the number $e$ can be expressed as $$\lim_{n\to\infty}(1 + 1/n)^n;$$ however, ...
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1answer
38 views

Sum of absolute values is finite

Suppose $\lambda_m \in \mathbb{R}$ and suppose that $\sum_{m \in \mathbb{N}} \lvert i + \lambda_m \rvert^{-p} < \infty$. Then why does $$\sum_{(m,n) \in\mathbb{N}\times\mathbb{Z}} \left\lvert i \pm ...
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0answers
23 views

Estimating a sum

i want to show the following: Assume that $\sum_{m\in\mathbb{N}}{|i+\lambda_m|^{-p}}<\infty$ (where $(\lambda_m)_m$ is a sequence of real numbers). I want to show that then also holds for each ...
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1answer
47 views

How we can show that $\:I_n\ge \frac{2}{\pi }\left(\frac{1}{n+1}+\frac{1}{n+2}+…+\frac{1}{2n}\right)$

We have $I_n=\int _{\pi }^{2\pi }\:\frac{\left|sin\left(nx\right)\right|}{x}\:dx,$ and we need to show that$\:I_n\ge \frac{2}{\pi }\left(\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}\right)$ I write ...
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2answers
53 views

radius of convergence of $\sum_{n=1}^\infty n!^2x^{n^2}$ [on hold]

Determine the radius of convergence of the following power series: $\sum_{n=1}^\infty n!^2x^{n^2}$
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1answer
39 views

Infinite summation of a trigonometric series

$\sum_{n=1}^{n=\infty}\sin(\frac{n\pi x}{L})\sin(\frac{n\pi y}{L})\surd(k^2+\frac{n^2 \pi^2}{L^2})$ I am trying to solve the above summation. I still could not figure out if this summation converges ...
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1answer
61 views

Simplifying Sum

How would one show that $$ \sum_{i=0}^n\binom{n}{i}(-1)^i\frac{1}{m+i+1}=\frac{n!m!}{(n+m+1)!} ? $$ Any hint would be appreciated. Note: I tried to recognize some known formula, but since I don't ...
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0answers
19 views

Help with simplification rules form sums and integrals.

IF you had a power series with summation notation and an integral what expressions would you be able to pull outside the integral and which would you be able to pull outside the sum.
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1answer
411 views

Simplification of a double sum involving partial sums of harmonic series

Could somebody explain the jump in the following equation? $$\frac{1}{n}\sum\limits_{i=1}^{n}\left[1 + \sum\limits_{j=i+1}^{n}\frac{1}{m}\right] = 1 + \frac{1}{nm}\sum\limits_{i=1}^{n}(n-i) $$
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0answers
31 views

Sum of combinatorics

I encounter the following sum of combinatorics and I can not seem to find the solution. Can anyone lend a hand? $\binom {n^c} {a} \binom {n^{c'}}{2a} + \binom {n^c} {a+1} \binom ...
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5answers
89 views

How to sum $\sum_{k=1}^n (k+1)(k)(k-1)$

Is there an intelligent way to do this sum without using sums of cubes and sums of squares? $$\sum_{k=1}^n (k+1)(k)(k-1)$$
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2answers
34 views

Taking the square of a finite series

I was reading something that involved taking the square of a sum $\sum_{i=1}^k(n_i-1)$. The author just presented the result. $$\left(\sum_{i=1}^k(n_i-1)\right)^2 = \sum_{i=1}^k(n_i^2-2n_i)+k + cross ...
5
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1answer
87 views

Find the value of $\sum_{m=1}^\infty tan ^ {-1}\frac{2m}{m^4+m^2+2}$

How to find value of this sum? $$\sum\limits_{m=1}^\infty \tan^{-1}\left(\frac{2m}{m^4+m^2+2}\right)$$ I can't understand how to simplify this. Should I use any trigonometric substitution to simplify ...
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1answer
25 views

Finding the radius of convergence for a Maclaurin series

I am required to find the radius of convergence for the function $$f(x) = 5x^3 - 6x^2 - 7x + 6$$ by first finding its Maclaurin series. I found the Maclaurin series to be $$ 6 - 7x -6x^2 + ...
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2answers
40 views

How to prove that a sequence converges

I am having some trouble understanding how I can show that a given series converges. I found a general explanation here that states: To prove that a sequence converges, it is sometimes easier to ...
3
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1answer
23 views

How to write product as a nested sums

I'm required to write $\prod\limits_{i=1}^n(1-j_{i})$ as a nested sum, where $j_{ll} \neq j_{k}$ if $u \neq k$. I undestand I'd get something in the form of ...
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2answers
32 views

Evaluate nested summation of a function

I'm trying to relearn summation simplification, I haven't touch math in a while. I'm having trouble simplify this nested summation down and I don't even know where to start. Could anyone please give ...
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1answer
64 views

How to show this equality holds?

I need to show that the following equality holds for any integer i,j,m,n and p where p is probability (0<=p<=1) Could you please help me?I think I should use hyper-geometric function but I could ...
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0answers
25 views

Series Expansion from Polynomial w/ Coefficients [on hold]

I have four coefficients to a 4-the order polynomial. Besides having some stroke of luck finding a pattern (that would be difficult considering the coefficient values) what is the best way to approach ...
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0answers
18 views

Explain the following summation

Suppose $K=(k_{d-1}, \dots\dots,k_{1},k_{0})_{2^w}$ then $KP = \displaystyle\sum_{i=0}^{d-1}(2^{wi}P)k_i = \displaystyle\sum_{j=0}^{2^{w}-1}\bigg(j\displaystyle\sum_{i:k_i=j}2^{wi}P\bigg)$ After ...
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2answers
43 views

Help me understand this algorithm problem.

First, I'm not looking for an answer here, I'm just looking to understand the problem so that I can prove it. I'm trying to analyzing the worst case running time of an algorithm, and it must has ...
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2answers
36 views

Reciprocal over a summation [closed]

Is this statement true? Can we take reciprocal over a summation? $$\frac 1{\sum_{n=1}^\infty\frac 1{(n+1)^3}}=\sum_{n=1}^\infty (n+1)^3$$
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0answers
44 views

how would i simplify this into an identity?

$$ B_{n,k}^{f\ln(g)} = B_{n,k}\left(\frac{d}{dx}[f(x)\ln(g(x))], \frac{d^2}{dx^2}[f(x) \ln(g(x)), \cdots, \frac{d^{n-k+1}}{dx^{n-k+1}}[f(x) \ln(g(x))]\right) $$ We know that: $$ B_{n,k}^{f\ln(g)} = ...
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0answers
24 views

Simplification of an expression sums and products

I am trying to simply the following expression: $ \sum_{\substack{\overline{t}_i , \overline{t}_j \in \{H,L\} \\ \forall j \in N_i}}^{} \sum_{j \in N_i}^{} \alpha_{\overline{t}_i , \overline{t}_{j}} ...
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2answers
70 views

Need to show following equality

I want to show that the following equality holds for any integer i,m, and n.I could not figure out how to show it analytically. Could you please help me? $$ \sum _{j=0}^n ...
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1answer
42 views

How do I extrapolate summation notation from a given series?

I am currently working on the power series for a homework assignment. I have to find the radius of convergence for the function $$\frac{10}{1+64x^2}$$ By setting up the $$\frac{1}{1+64x^2}$$ part ...
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1answer
22 views

Summation simplification explanation

I'm trying to understand summation for my algorithm course and it has been a while since I took discrete math. Could any body please explain how does summation simplification work from the problem ...
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1answer
51 views

How to choose a contour in order to use the residue theorem to sum up a series from Ryzhik?

I would like to know how to sum up to following series (from the Gradshteyn-Ryzhik tables): ...
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3answers
39 views

Finding the formula for summation of the series

I was just solving a competitive programming question, wherein I found out that a formula can be used for solving it efficiently. Problem statement: http://www.spoj.com/problems/TOHU/ I tried a lot to ...
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4answers
169 views

Calculate the infinite sum $\sum_{1}^\infty \frac{\log{n}}{2n-1}$

I would like to calculate an asymptotic expansion for the following infinite sum: $$\displaystyle \sum_{1}^N \frac{\log{n}}{2n-1}$$ when $N$ tends to $\infty$. I found that the asymptotic expansion ...
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1answer
43 views

Why the sum of the list is 4?

Wolfram Alpha says Sum[Sin(Pi*n/4)]/(Pi*n/4),{n,-Infinity,Infinity}] is equal to 4 but I don't know how to resolve it... In my signal and system homework,this ...
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1answer
55 views

Sum of Number of non-decreasing sequences [duplicate]

I know that the number of non-decreasing sequences of length $n$ and numbers in the sequence lying in the range $[l,r]$ is given by $$\binom{n+r-l}{n}$$ What is the formula to find the ...
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0answers
42 views

Is there a closed-form expression for the following sum?

Is there a closed-form expression for the following sum: $$\large\sum_{\{n_i\}} \frac{x_i^{n_i}}{\prod_i (i!)^{n_i} n_i!}$$ where the sum runs over all combinations of $\{ n_{i=0,\dots,k} \}$ such ...
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2answers
119 views

Number of subsets of length 7 [duplicate]

I have the following summation: $$\sum\limits_{k=7}^{n} {k-1\choose 6} $$ and apparently it counts the number of subsets of {1, 2, . . . , n} having size 7. Why is this?
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1answer
49 views

Summation giving incorrect value (I think) in Mathematica

I have the problem $\displaystyle\sum\limits_{x=2}^4 \dfrac{1}{x^2}$. I know the answer to this problem should be $\dfrac{61}{144}$. When I type the problem into Wolfram Mathematica, I get the answer ...
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2answers
45 views

Solving inequality equation involving sum of binomial coefficients

I have a function $f(k,\,i)$ involving binomial coefficients: $$f(k,\,i)\,=\left(\begin{matrix}k+i \\ k\end{matrix}\right)=\frac{(k+i)!}{k!\,i!}$$ And the following sum over this function (expansion ...
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1answer
29 views

Axiom of Induction Question

I have to use the axiom of induction to prove the summation of k^3 from 0 to n is $(n(n+1)/2)^2$. Here's what I have so far: Let P(n) be the assertion that $0^3+1^3+⋯+n^3=(n(n+1)/2)^2$ Base Case ...
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2answers
62 views

Number addition riddle

I got this math "riddle" in one of my math test, and I would love to know how to solve it. If $$S = 1 + 2 + 3 + 4 + \ldots + 2015,$$ then a sum of $$1 + 2 + 3 + \ldots + 2015 + 2016 + \ldots + 4030$$ ...
0
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1answer
33 views

Do the values above and below a Sigma in a summation have a name?

As the title asks: do the numbers above and below a sigma have a specific technical name? I am trying to describe an inefficiency in an algorithm, where the set of items used in the summation could ...