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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2answers
37 views

Double sum of products of integers up to $n$

Suppose that $S$ is defined by $$ S(n) = \sum_{i=0}^{n} \sum_{j=0}^{i} ij. $$ I'm confused as to how $S(3) = 25$ from this summation. Can anyone expand on it as to how to get the answer?
0
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0answers
25 views

Matsubara sum with general exponent

Matsubara sums of the form $$\sum_{i\omega}\frac{1}{(i\omega-\xi_1)^a}\frac{1}{(i\omega-\xi_2)^a} $$ have closed-form solutions for $a=1,2$. See Wikipedia. Are there also closed-form solutions for ...
0
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0answers
12 views

Simplifying an expression which includes summation symbols and the cumulative distribution function for the normal

I would like to be able to simplify the expression: $E(Y|\mu,\sigma^2) = \frac{\sum_1^J 1 - 2 \Phi((c_j - \mu)/\sigma) + 2 \Phi((c_j - \mu)/\sigma)^2}{J}$ where $\Phi$ is the cumulative distribution ...
2
votes
0answers
52 views

Evaluate $\sum_{n=1}^\infty\frac{2^{-2^n}\cos{(2^n)-2^{-3(2^n)}\cos{(3(2^n))}}}{2^{2^{n+2}}-2^{1-2^{n+1}}\cos{(2^{n+1})}+2}$

I want to find the value of $$\sum_{n=1}^\infty\frac{2^{-2^n}\cos{(2^n)-2^{-3\cdot2^n}\cos{(3\cdot2^n)}}}{2^{2^{n+2}}-2^{1-2^{n+1}}\cos{(2^{n+1})}+2}$$ We have an identity ...
7
votes
1answer
112 views

$S1 = 1 + {x^3 \over 3!} + {x^6 \over 6!} + …$

In one of my lecturer's problem sheets we were asked to evaluate the following sums: $$S1 = 1 + {x^3 \over 3!} + {x^6 \over 6!} + \dots $$ $$S2 = {x^1 \over 1!} +{x^4 \over 4!} +{x^7 \over 7!} + ...
1
vote
0answers
30 views

How does one generally express a symmetric summation into matrix multiplication?

In the summation ,$$\sum_{i}^{}\sum_{j}A_{ij}X_{i}X_{j}$$ a nice symmetry exists. The final sum of this summation is just \begin{matrix} (A_{11} & A_{12} & A_{13}) X_{1} \\ (A_{21} & ...
1
vote
0answers
27 views

Binomial square sum and product

Given $c,n\in\Bbb N$ what is the expression for $$S(n,c)=\binom{n}c^2+\binom{n-c}c^2+\dots+\binom{x}c^2$$ and $$P(n,c)=\binom{n}c^2\cdot\binom{n-c}c^2\cdot\dots\cdot\binom{x}c^2$$ where $x-c<c\leq ...
2
votes
1answer
29 views

Einstein Summation with Del Operator

Can someone show explicitly me why $2B_k\nabla B_k = \nabla B^2$ ? Is $B_k\nabla B_k$ just $B_x\nabla B_x+B_y\nabla B_y+B_z\nabla B_z$? But then I end up with nine terms on the LHS and I can't ...
1
vote
2answers
44 views

I Know that $\sum_{n=0}^\infty \frac{1}{n}$ Diverges, but what is $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ [duplicate]

We know that $\sum_{n=0}^\infty \frac{1}{n}$ diverges since it is a harmonic series. However, I was recently working on a homework problem where I was given to find if $\sum_{n=1}^{\infty} ...
3
votes
2answers
75 views

Is there a name for a binomial expansion without coefficients?

I am investigating a problem from George E. Andrews Number Theory (Dover, 1971), discussed previously here: Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$ I was led ...
2
votes
1answer
25 views

Assymptotics of the generalized harmonic number $H_{n,r}$ for $r < 1$

The $H_{n,r}$ generalized harmonic number is defined as: $$H_{n,r} = \sum_{k=1}^{n} \frac{1}{k^r}$$ I'm interested in the growth of $H_{n,r}$ as a function of $n$, for a fixed $r\in[0,1]$. For ...
4
votes
1answer
134 views

Tricky proof that the weighted average is a better estimate than the un-weighted average:

The following is a word for word copy of a tough question and the solution to it. I have marked $\color{red}{\mathrm{red}}$ the parts of the solution for which I do not understand and the parts marked ...
1
vote
0answers
9 views

Convergence of a Double Summation solution to Laplace's Equation

For a cube of side length $a$ with 2 opposite sides held at the same potential $V$, the potential at the center of the cube can be expressed in series form as And I am trying to show that this ...
3
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1answer
59 views

$H_k$ summability of a sequence implies its Abel summability to the same sum.

Let $\sigma_n^{(k)}=\frac{1}{n+1}\sum_{j=0}^{n}\sigma_j^{(k-1)}$ and $\sigma_n^{(1)}=\frac{1}{n+1}\sum_{j=0}^{n}s_j.$ If $\lim_{n\to \infty}\sigma_n^{(k)}=L$ we call the sequence $(s_n)$ is summable ...
3
votes
2answers
73 views

Prove: $\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=1}^{n}\sqrt{1+\frac{k}{n}}=\frac{2}{3}(2\sqrt{2}-1)$

Prove: $\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=1}^{n}\sqrt{1+\frac{k}{n}}=\frac{2}{3}(2\sqrt{2}-1)$ What method to use in order to find the closed form of summation ...
2
votes
2answers
41 views

How to recognize / convert a tricky limit of an infinite series as a Riemann integral?

Edit: I've modified the sums and integrals below into convergent sums and integrals, but my questions are still the same - how can I convert sums into integrals legitimately? As far as I know, the ...
0
votes
1answer
27 views

How to write sum notation for an array of 2D points

What is a correct way to write using sigma notation for a problem involving an array of 2-dimensional points. Say I have 2 arrays, $P_{e}$ and $P_{a}$, both containing $N$ elements. $P_{e}$ represents ...
9
votes
4answers
141 views

What does the false infinite sum of a series mean?

For any geometric series with |$r$| < 1 , I know that $$\sum_{k=1}^{∞} ar^{k-1} =\frac{a}{1-r}$$ But if |$r$| > 1 and you try to use the formula, you'll get a weird answer. For instance: ...
2
votes
1answer
93 views

Summation relating factorial and cosine

How to simplify \begin{align*} \sum_{k=0}^{\infty}\left(-1\right)^{k}\frac{\left(2k\right)!}{4^{k}\left(k!\right)^{2}}\cos\left(kx\right) \end{align*} for $0\leq x <\pi$ ? I don't even know where ...
19
votes
3answers
427 views

Mysterious identity

Playing around with Maple I found this identity $$\sum_{k=0}^{n-1}\frac{2k+1}{1-z^{2k+1}}=n\sum_{k=0}^{n-1}\frac{1}{1+z^{k}}$$ where $n$ is a positive integer, $z=\exp(\pi i/n)$. I was able to verify ...
0
votes
2answers
49 views

Show that $\sum_{n=0}^\infty\frac{(-1)^n}{(2 n+1) \left(\frac{x^2}{(2 n+1)^2}+1\right)}=\frac{\pi}{4\cosh\left(\frac{\pi x}{2}\right)}$

Is it true that $$\sum_{n=0}^\infty\frac{(-1)^n}{(2 n+1) \left(\frac{x^2}{(2 n+1)^2}+1\right)}=\frac{\pi}{4\cosh\left(\frac{\pi x}{2}\right)}$$? How can I prove it ? Thanks for helping.
2
votes
0answers
72 views

Evaluate $\sum_{n=0}^{18}\sin{\left(\frac{(2n+1)^2\pi}{38}\right)}$ and $\sum_{n=0}^{18}\sin{\left(\frac{(2n)^2\pi}{38}\right)}$

How to show that $$\sum_{n=0}^{18}\sin{\left(\frac{(2n+1)^2\pi}{38}\right)}=\sum_{n=0}^{18}\sin{\left(\frac{(2n+1)^4\pi}{38}\right)}=\sum_{n=0}^{18}\sin{\left(\frac{(2n+1)^8\pi}{38}\right)}=0$$ and ...
1
vote
0answers
30 views

Dividing Two Sums

I have three sets of data points, let's say...$\{1,2\}\cup\{3,4\}\cup\{5,6\}$. Now suppose that I want to sum up the first point in each, and divide it by the sum of the quotients of each pair, e.g. ...
2
votes
3answers
56 views

Limit of a Sum as $n \to \infty$ [closed]

I'm trying to evaluate this problem. The answer is $e^2 - e$, but I can't seem to get to that answer. $$\lim_{n\to\infty} \frac{1}{n} \left ( \sum_{i=1}^{n} e^{i/n} \right )$$
0
votes
1answer
24 views

Series with parametric value

I have some problem with this parametric series: $ \sum_{i=1}^\infty n^\alpha\left(\frac 1{n^{1/4}}-\sin\left(\frac 1{n^{1/4}}\right)\right) $ which value of $\alpha$ makes the series convergent? And ...
4
votes
1answer
59 views

How to evaluate $\sum_{n=1}^m 2^n \arctan 2^n \theta$

I need to evaluate $$\sum_{n=1}^m 2^n \arctan 2^n \theta$$ as a function of $m$ and $\theta$. All I've done so far is write out the series explicitly: $$\sum_{n=1}^m 2^n \arctan 2^n \theta = 2 ...
4
votes
1answer
77 views

Summation of fractional parts $\frac{m}{n}$, where $2 \leq n < m$ (amateur)

I am looking for the result of the sum of the fractional part of the following number: $$f(m):=\sum_{n=2}^{m-1}Frac\left(\frac{m}{n}\right)$$ After some research I have found $2$ possible ...
5
votes
4answers
637 views

How can I calculate or at least approximate the sum?

As a part of a complexity analysis of the algorithm, I have to calculate the following sum: $$n^{1/2} + n^{3/4} + n^{7/8} + ...$$ where in total I have $k$ elements to sum: ...
0
votes
2answers
58 views

Why is this summation equal to …

I have this equation $\sum_{j=0}^{\infty }j(j+1)c_{j+1}\varrho ^{j}+2(L+1)\sum_{j=0}^{\infty }(j+1)c_{j+1}\varrho ^{j}-2\sum_{j=0}^{\infty }jc_{j}\varrho ^{j} +(\varrho_0-2(L+1))\sum_{j=0}^{\infty ...
0
votes
1answer
13 views

Showing that the negative binomial PMF is valid

Let $X\sim\text{NegBin}(m,p)$ where $P[X=n]={n-1\choose m-1}p^m(1-p)^{n-m}$ for $n\geq m$. I want to verify that $$\sum_{n=m}^\infty{n-1\choose m-1}p^m(1-p)^{n-m}=1$$ but it's not easy to see how I ...
2
votes
0answers
26 views

How to evaluate a sum of rational expression containing $\sin$?

How one can find the value of the series $$\sum_{n=1}^\infty \frac{8\sin((2n-1)x)}{\pi (4(2n-1)-(2n-1)^3)}?$$ Wolfram $\alpha$ gave an ugly expression involving $i$ but I think one can simplify the ...
2
votes
2answers
38 views

Why does $\int_{-L}^{L} \sum_{n=1}^{\infty}a_n\cos \frac{n\pi x}{L}=\sum_{n=1}^{\infty}a_n\int_{-L}^{L}\cos \frac{n\pi x}{L}$

Why does $$\int_{-L}^{L} \sum_{n=1}^{\infty}a_n\cos \frac{n\pi x}{L}=\sum_{n=1}^{\infty}a_n\int_{-L}^{L}\cos \frac{n\pi x}{L}$$ This is used in a derivation of the Fourier coefficients. I see why ...
3
votes
3answers
99 views

Calculating the 12 days of christmas by hand

For an exercise in my math class we are calculating the cost of the 12 days of christmas. Let's define a set $c$ to be the price of each item in the popular "12 days of christmas" song, from a ...
1
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2answers
23 views

Formulating the bitwise OR operation

Considering the bitwise OR operation, wikipedia states, $$x\;\mathrm{OR}\;y = \sum_{n=0}^b ...
6
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0answers
149 views

How do i evaluate this sum $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2n!}$?

How do I evaluate this sum: $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2n!}$$ Note: The series converges by the ratio test. I have tried to use this sum:$$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}= ...
1
vote
3answers
31 views

Summation of a constant using sigma notation

Apologies if this is a silly question, but is it possible to prove that $$\sum_{n=1}^{N}c=N\cdot c$$ or does this simply follow from the definition of sigma notation? I am fairly sure it's the ...
-1
votes
1answer
77 views

What is the value of the series? [closed]

What is the value of the summation $$\dfrac{1}{1!} + \dfrac{1+2}{2!} + \dfrac{1+2+3}{3!} + \dots + {{1+2+3+\dots+i}\over{i!}} + \dots $$ The sum is till infinity.
-1
votes
1answer
27 views

compound interest with monthly deposit

$150 is deposited into an account at beginning of each month that pays 6% compounded monthly. What is the account's value after 10 years? I know how to find the sum using regular compound interest, ...
0
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2answers
42 views

Simple summation problem regarding origin of summand:

If $$\frac{1}{\sigma_\widehat{e}^2}=\sum_i\frac{1}{\sigma_i^2}\tag{1}$$ Pick any one of the $\sigma_j$ and multiply both sides of $(1)$ by $\sigma_j^2$ $$\implies\frac{\sigma_j^2}{\sigma_\widehat ...
0
votes
1answer
23 views

General Notation for a Reductive Operation, such as Sum (Σ) or Product (Π)

In functional programming, people often use operations like "fold" or "reduce", to convert from a collection to a single object using a binary operation. This is analogous to the sum and product ...
2
votes
3answers
66 views

Bounds on Gaussian infinite sum

What are some good upper and lower bounds on the following sum? $$S=\sum_{n=-\infty}^{+\infty}\dfrac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{1}{2}\left(\frac{n}{\sigma}\right)^2}$$ I am looking for ...
2
votes
1answer
28 views

Show that the moment generating function of $ W$ is $M_W(t) = (qe^t+p)^n$

If $Y$ is a random variable with moment-generating function $M_Y(t)$ and if $W$ is given by $W=aY+b$, then the moment generating function of $W$ is $e^{tb}M_Y(at)$ Suppose that $Y$ is a binomial ...
1
vote
0answers
36 views

Is this sum zero?

Let $\{y,...,x\}$ be a finite subset of the natural numbers. Does this imply that $$\sum_{l=y}^{x} \prod_{ p \in \{y,...,x\} \backslash\{q\}} \prod_{q \in \{y,...,x\} \backslash \{l\}} ...
1
vote
2answers
80 views

Divergence of a positive series over a uncountable set. [duplicate]

Let $\Lambda$ be a uncountable set and let $\{a_{\alpha}\}_{\alpha\in\Lambda}$ be such that $a_{\alpha}>0$ for all $\alpha \in\Lambda$ proof that, $$\sum_{\alpha\in\Lambda}a_{\alpha}$$ ...
1
vote
1answer
59 views

Infinite trigonometric summation

Can the following summation be written in a finite number of terms: $$\sum_{r=1}^{\infty}\frac{\tan(\theta/2^r)}{2^{r-1}\cos(\theta/2^{r-1})}$$ I tried to simplify the expression using trigonometric ...
1
vote
0answers
23 views

Combinations and sums

In how many ways can we get a sum greater than $x$ for $n$ distinct integers between $1$ and $x-1$, both inclusive? For example For $x = 5$ and $n = 3$, the required combinations are $(1,2,3)$, ...
0
votes
1answer
42 views

Is there a means of finding an infinite sum by means of altering it into an integral?

If you are given a sum, say $$\sum_a^b f(x)$$ with $a,b\lt \infty$ Is there a means of solving for this sum by means of integration? (I am familiar with sophomores dream.) Thank you for any help
1
vote
1answer
52 views

how to minimize a summation containing an absolute value

Thank you all in advance I have been having some trouble figure out the following problem. You are given a sample {$y_i$}, i=1,… N, from an unknown probability distribution p(y). I want to show the ...
2
votes
1answer
39 views

Alternative factorization of $\prod\limits^{n}_{k=1}k!^{k+1}$

Question: How can I succinctly express (using the product and sum notations) the following expression? $$n^{(n+1)}(n-1)^{(n+1)+n}(n-2)^{(n+1)+n+(n-1)}\cdot\cdot\cdot ...
0
votes
0answers
20 views

Summation notation with a comma- what does this mean

I am reading a document with this expression: evidencei=∑jWi, jxj+bi Can someone explain what this means? In context: We also add some extra evidence called a bias. Basically, we want to be ...