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

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0
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1answer
21 views

How do we approach this summation question?

How do we find the following sum--- $\displaystyle\sum_{m=1}^\infty \displaystyle\sum_{n=1}^\infty \frac{mn^2}{2^n(n2^m+m2^n)}$
0
votes
1answer
6 views

How to show Thomas-Reiche-Kuhn sum rule

Given: $$ f_{ni} = \frac{2m\omega_{ni}}{\hbar}\big|\langle n | x |i\rangle\big|^2, $$ and a Hamiltonian in the form $$ H_0=\frac{p^2}{2m}+V(x), $$ I would like to show the following sum rule (known as ...
3
votes
1answer
29 views

Can a general formula be developed for these integrals?

Can the following integral $I$ be written wrt $n$? $$I=\displaystyle\int_{0}^{\infty} \dfrac{dx}{\sum_{k=0}^{n} x^k}$$ I found the values for $n=1,2,3$ but can we generalize it for an arbitrary ...
5
votes
4answers
197 views

Why is $\int\limits_{1}^{n} \log x \,dx \le \sum\limits_{x = 1}^{n}\log x$?

It has been a long time since I studied integrals, so this question may sound stupid. I was going through this wiki page, and came across the following inequality: $$\int_{1}^{n} \log x \,dx \le ...
1
vote
1answer
14 views

Easy way to compute $Pr[\sum_{i=1}^t X_i \geq z]$

We have a set of $t$ independent random variables $X_i \sim Bin(n_i, p_i)$. We know that $$Pr[X_i \geq z] = \sum_{j=z}^{\infty} { n_i \choose j } p_i^j (1-p_i)^{n_i -j}.$$ But is there an easy way to ...
0
votes
1answer
44 views

Limit where wolfram alpha does not help

With the help of Wolfram Alpha I got the following $$\frac{\sum_{i=x+1}^{\infty} i q^i}{\sum_{i=1}^x i q^i}=\frac{q^x (x(1-q))+1}{q^x (x(q-1)-1)+1}$$ Now I am intersted in the behavior of this sum ...
0
votes
3answers
42 views

Closed-form term for $\sum_{i=1}^{\infty} i q^i$ [duplicate]

I am interested in the following sum $$\sum_{i=1}^{\infty} i q^i$$ for some $q<1$. Is there a closed-form-term for this? If yes, how does one derive this? I am also interested in ...
0
votes
1answer
36 views

Summing dependent random variables with unknown joint cdf

Suppose that X_1, X_2,... X_5000 are discrete and dependent non-identically distributed random variables, whose marginal distributions are known, but whose joint distribution is not known. Is there ...
0
votes
1answer
90 views

How does one graph $\sum_{x=0}^{n}$ [on hold]

How does one graph a summation, like $$\sum_{x=0}^{n} n$$ Can it be like this Because if you take the points from the summation (0,0), (1,1), (2,3), (3,6) you can tell by summations it only works ...
0
votes
0answers
18 views

Sum of squares of series of boolean variables

I am going to simplify the following series: $$\sum^4_{v=1} \left(1 - \sum^4_{i=1} x_{v,i}\right)^2 + \sum^4_{i=1} \left(1 - \sum^4_{v=1} x_{v,i}\right)^2$$ Since $x_{i,j}$ is a boolean variable, ...
0
votes
1answer
49 views

Sumatory formula

Anybody knows the formula for this, because I don't know how to write it from the basic formula of $$\frac{n(n+1)}{2}$$: $$\sum _{i=1}^{n}{ \sum _{j=1}^{ n}{ \sum _{ k=1 }^{ n }{ \sum _{ h=1 }^{ n ...
0
votes
0answers
21 views

Simplification of a power weighted alternating binomial sum

Given positive integers $T$, $n$ and $m$ and real number $p$ with $0< p < 1$, how can I simplify the following binomial sum: $$ \sum_{k=m}^{\lfloor ...
1
vote
0answers
36 views

Sum of product of binomial coefficients and exponential function

I would like to know how to obtain (if it exists) a closed form expression of the sum $$S=\sum^{n}_{k=0}2^k{{n+1}\choose k}{{r-n-2}\choose {n-k}}$$ So far, I have tried to use the method of ...
3
votes
2answers
1k views

Sigma notation only for odd iterations

$ \sum_{i=0}^{5}{i^2} = 0^2+1^2+2^2+3^2+4^2+5^2 = 55 $ How to write this Sigma notation only for odd numbers: $ 1^2+3^2+5^2 = 35 $ ?
12
votes
3answers
202 views

How to prove that $ \sum_{n=1}^{\infty}n\prod_{k=1}^{n}\frac{1}{1+ka} = \frac{1}{a} $?

Mathematica tells me that $$ \sum_{n=1}^{\infty}n\prod_{k=1}^{n}\frac{1}{1+ka} = \frac{1}{a} $$ I could prove it for $a\rightarrow 0$, $a=1$ and $a\rightarrow \infty$, but could not find a general ...
0
votes
2answers
31 views

Sum of this series.

I tried manipulating it to get it into a binomial expansion of two known terms, but i seemingly failed. Please help me out. $$S=\displaystyle\sum_{r=0}^{12} \binom{12}{r} \cos \frac {r\pi}{6}$$
1
vote
1answer
55 views

Finding formulas for sums

I know that $\sum_{d \mid n} \mu(d) = 0$ whenever $n >1$, and I know that $\sum_{d \mid n} \phi(d) = n$. How can I use this in order to give a formula for $\sum_{d \mid n} \mu(d)\phi(d)$?
0
votes
1answer
87 views

Big Mathematics Challenge on Set and Summation? [on hold]

please be aware that this is not homework. it's past PHD entrance Exam on 2011. Suppose: $$B=\{(A_1,A_2,A_3) \mid \forall i; 1\le i \le 3; A_i \subseteq \{1,\ldots,20\}\}$$ if we have: ...
0
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0answers
11 views

Weighted sum VS weighted sum of harmonic mean

Suppose we have a vector: A, with each element between 0 and 1. The Harmonic mean of A is H. And we have a positive weight vector: W, W(i)>0. I want to compare sum(W.*A) and sum(W*H) I tried all ...
2
votes
4answers
104 views

Fastest way to integrate $\int_0^1 x^{2}\sqrt{x+x^2} \hspace{2mm}dx $

This integral looks simple, but it appears that its not so. All Ideas are welcome, no Idea is bad, it may not work in this problem, but may be useful in some other case some other day ! :)
1
vote
1answer
33 views

defenite integral involve bessel function

I have an integral which involves Bessel function as follows: $I=\int_{r=0}^a \int_{\theta=0}^{2\pi}(e^{-jkr\cos(\theta-\phi)}d\theta)rdr$ I have tried with $e^{-jkr\cos(\theta-\phi)}=\sum ...
2
votes
1answer
33 views

What is the inverse function of $\int{ \frac{1}{{\sqrt{x+1}}{x^n}} dx}$?

I am trying to solve $$ \frac{dy}{dt} = \alpha ((y+1)^2 - \gamma)^n \hspace{2cm} y(0)=0 $$ Here $y$ is a real-valued, monotonically increasing, positive definite function of $t$ in the interval ...
1
vote
4answers
71 views

closed-form term for this sum:

related to this question: Is there an easy closed-form term for $$\sum_{j=k}^{\infty} \frac{x^j}{j!}e^{-x},$$ thus when the sum starts at a constant $k$ instead of $1$? EDIT: Thanks for your help. ...
4
votes
6answers
824 views

Calculate $\sum_{k=1}^n \frac 1 {(k+1)(k+2)}$

I have homework questions to calculate infinity sum, and when I write it into wolfram, it knows to calculate partial sum... So... How can I calculate this: $$\sum_{k=1}^n \frac 1 {(k+1)(k+2)}$$
2
votes
2answers
82 views

How find this sum $S(x)=\sum_{k=1}^{\infty}\frac{\cos{(2kx\pi)}}{k}$

Find this sum $$S(x)=\sum_{k=1}^{\infty}\dfrac{\cos{(2kx\pi)}}{k},x\in R$$ my idea: since $$S'(x)=2x\pi\cdot\sum_{k=1}^{\infty}\sin{(2kx\pi)}$$ then I can't.
5
votes
2answers
104 views

How find this $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}\zeta_{n}(3)}{n}=?$

Question: show that $$\sum_{n=1}^{\infty}\dfrac{(-1)^{n-1}\zeta_{n}(3)}{n}=\dfrac{19\pi^4}{1440}-\dfrac{3}{4}\zeta{(3)}\ln{2}?$$ where $$\zeta_{n}(3)=\sum_{k=1}^{n}\dfrac{1}{k^3}$$ But I use ...
0
votes
1answer
22 views

Finding final value of sum from double summation code fragment.

sum = 0; inc = 0 for i from 1 to n for j from 1 to i sum = sum + inc inc = inc + 1 What does it mean when it is asking the "final value of sum" from above code ...
4
votes
2answers
69 views

Show that the following series is less than $4 \pi^2 / 3$.

Show: For any $k = 0,1,2,...$, $$ \sum_{i=0}^{i=k} \frac{(k+1)^2}{(i+1)^2 (k-i+1)^2} \leq \frac{4 \pi^2}{3}. $$
3
votes
2answers
26 views

Find value of $n$ with given conditions

The 4-digit positive number $n$'s digit sum is $20$. The sum of the first two digits is $11$, the sum of the first and the last digit as well. The first digit is the last digit $+3$. What is the ...
0
votes
1answer
23 views

Summing standard deviations? or summing variances?

I have a method for making measurements. Within this method there are four separate variables that can influence the measurement (i.e. variation). I have individually tested each variable (by making ...
1
vote
3answers
60 views

Can you show that the LHS equals the RHS in this equation, by showing how I can get the expression on the RHS?

$$ \frac{1^2+2^2+...+(n-1)^2}{n^3} = \frac{(n-1)n(2n-1)}{6n^3} $$ Can someone show me step by step how I can transform the LHS to the RHS? If possible, using high school-level math. I have now ...
-1
votes
1answer
38 views

Simple understanding of Sum symbol.

I wonder if this is correct to write even if nj has unik length for each variable, so for x it can be 12, but for y it can be 20 and for z it can be 40. Also is it possible to convert it to ...
-1
votes
2answers
53 views

How to approximate this large sum of exponential terms

Is there any way to approximate the following sum: $$ \sum\limits_{i_1=1}^N\sum\limits_{i_1=2}^N \cdots \sum\limits_{i_k=1}^N \cdots\sum\limits_{i_N=1}^N \exp(-r_{i_1}-r_{i_{k+1}}-r_{i_{2k+1}}- ...
2
votes
5answers
220 views

Easy summation question [duplicate]

While during physics I encountered a sum I couldn't evaluate: $$S= 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}\cdots$$ Is there a particular formula for this sum and does it converges?
4
votes
2answers
342 views

n more each day

It's been a while since I've been at school and I don't work in a field that practices this sort of stuff so I don't know the formula my brain can't wrap my head around the problem. The problem: You ...
12
votes
6answers
826 views

Proving a binomial sum identity

Mathematica tells me that $$\sum _{k=0}^n { n \choose k} \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}.$$ Although I have not been able to come up with a proof. Proofs, hints, or references are all ...
0
votes
1answer
39 views

I can't get the answer of this problem about radius of convergence

I have this one: $$ \sum_{n=1}^{\infty} {\frac{\ln{(3n^{2}+5)}x^{n}}{n^2 - 3n +5}} $$ I tried with the classic method: $$ \sum_{n=1}^{\infty} {\frac{\ln{(3n^{2}+5)}((n+1)^2 -3(n+1) + 5)x^{n}}{(n^2 ...
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vote
3answers
32 views

Compute radius of convergence

Someone can help me with this radius of convergence please? $$ \sum_{n=0}^{\infty}{\frac{3^{2}(5n^{7}+2)}{2^{n}(5n^{3}-1)}}x^{n} $$ I tried $$ r = \lim_{n \to \infty } ...
0
votes
3answers
42 views

Compute this radius of convergence

Someone can help me with this radius of convergence please? $$ \sum_{n=0}^{\infty}{n^{1/n}x^{n}} $$ I tried $$ r = \lim_{n \to \infty } \frac{n^{\frac{1}{n}}x^{n}}{(n+1)^{\frac{1}{n+1}}(x^{n+1})} ...
1
vote
1answer
43 views

Is an integral without a differential component on a finite number of points just a sum?

Is an integral $$\int_{\lbrace 1, 2, 3 \rbrace} f(x)$$ simply the sum $$\sum_{x=1,2,3} f(x)?$$ I ask this question because of the generalization to multiple dimensions of integration by parts ...
0
votes
1answer
34 views

Comfirmation of third derivative of symbolic equation including summation

With previous help I was able to find the first derivative of an equation for a work project. Now I'm after the second and third derivative, for use in a program to find the maximum (Which I must do ...
0
votes
3answers
43 views

How do I finish this summations problem?

I have posted a picture since I don't know how to make the summation symbols with the lower and upper summations on keyboard, sorry about that.. $$\sum_{a=1}^9\sum_{b=0}^9(101a+10b)$$ The answer is ...
-1
votes
1answer
63 views

Rewrite and approximate the sum as an Integral $\sum_{i=1}^{1000} \sqrt{i}$ [closed]

This is not an Infinite sum !, how do we change this to an Integral. $ $ We normally write an integral as an infinite sum.
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vote
3answers
63 views

Find all values of $c$ for which the following series converges $\sum_{n=1}^{\infty} \left(\dfrac{c}{n}-\dfrac{1}{n+1}\right)$

I know that the series in question converges when $c=1$, but I have no concrete way to find all such values of $c$ for which this is true.
0
votes
0answers
43 views

Intuition behind summation [duplicate]

$$\sum_{n=1}^{\infty}n = -1/12$$ Could someone give me an intuitive explanation of why this is true? I just completed Calc 2 and finished a unit on series, and according to the stuff we learn't the ...
5
votes
1answer
61 views

How many decimal representations are possible for the number 1

I know that there at least two $0.\overline{9}$ and 1 Is there a neat and more concrete way to understand this problem.
11
votes
1answer
77 views

Convergence of $\sum_{n=1}^{+\infty}\frac{(-1)^{f(n)}}{n}$ where $f(n)$ is the number of prime divisors

Let $f(n)$ be the number of prime divisors of a number $n$ counted with their multiplicities. Show that the series $\sum_{n=1}^{+\infty}\frac{(-1)^{f(n)}}{n}$ converges and has sum $0$. Attempt ...
4
votes
2answers
72 views

How to evaluate the sum $\sum_{k = 0}^{n}2^k {{n}\choose {k}}$ [duplicate]

How do I evaluate the sum: $$\sum_{k = 0}^{n}2^k {{n}\choose {k}}$$ I know that $2^k = {n \choose 0} + {n \choose 1} + {n \choose 2} + {n \choose 3}... {n \choose k}$, but I don't know how to proceed ...
1
vote
1answer
114 views

Find the sum of the series $\sum \limits_{n=3}^{\infty} \dfrac{1}{n^5-5n^3+4n}$

Feel free to skip obvious steps, or use a calculator when required. I just want to understand the theme of the solution. Any help is appreciated EDIT : We can write$$ \dfrac{1}{n^5-5n^3+4n} = ...
1
vote
1answer
16 views

Proving an identity involving binomial coefficients and fractions

I've been trying to prove the following formula (for $n > 1$ natural, $a, b$ non-zero reals), but I don't know where to start. $$\sum_{j=1}^n \binom{n-1}{j-1} \left( \frac{a-j+1}{b-n+1} \right) ...