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Questions tagged [summation]

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

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6 views

Show that $(1+|y|^{2})^{2}((dy^{1})^{2}+\ldots+(dy^{n})^{2})$, where $|y|^{2}=\sum_{i=1}^{n}{(y^i)^2}$.

I need prove the following, Show that, $$\sum_{i=1}^{n}{\left((1+|y|^2)dy^{i}-y^{i}(2y^{1}dy^{1}+\ldots+2y^{n}dy^{n})\right)^{2}}+4(y^{1}dy^{1}+\ldots+y^{n}dy^{n})^{2}=(1+|y|^{2})^{2}((dy^{1})^{...
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1answer
9 views

The Lagrange Interpolation formula – Spivak's Calculus Ch 3 Problem 7(b)

The problem: Now find a polynomial function $f$ of degree $n - 1$ such that $f(x_i) = a_i$, where $a, \ldots, a_n$ are given numbers. I found that this question had been asked before, but I did not ...
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1answer
36 views

Find the Exact value of $\lim_{n\to\infty} \sum_{k=1}^n \int_0^{\pi} e^{-nx}\sin (kx) dx$

Find the Exact value of : $$\lim_{n\to\infty} \sum_{k=1}^n \int_0^{\pi} e^{-nx}\sin (kx) dx$$ What I did : Take $I=\int_0^{\pi}e^{-nx}\sin(kx)dx$ After Integration, I get $$I=\frac{k(1-e^{-n\pi}\...
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0answers
11 views

Integration of a function given by a partial sum and “o” notation

If $\sum_{n=1}^{\infty}a(n)=A$, then I know that $\sum_{n=1}^{N}a(n)=A\pm o(1)$, as $N\to\infty$. My question is about if it is ok the following equality, $$\int_{1}^{K}\sum_{n=1}^{[x]}a(n)\,f(x)\,...
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3answers
28 views

Binomial Theorem Inductive Proof - a reindexing moment

I'm copying a proof from someone else and they make this move I don't feel comfortable with. So in the inductive step we assume $ { \left( x+y \right) }^{ n }= \sum _{ m=0 }^{ n }{ \left( \begin{...
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0answers
25 views

Dealing with a polynomial sum

I am trying to approximate the following function, $$f(x)=\frac{\sum^{N}_{c=1}\gamma_c x^{c+k-2}-\sum^{N}_{c=1}\beta_c x^{c-1}}{\sum^{N}_{c=1}\alpha_c x^{c}}$$ where $\alpha_c$'s, $\beta_c$'s and $\...
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1answer
12 views

Most elegant way to show Inner-Product of tensors with indices?

How would you write this sum most elegantly? (Or in other words: What is the most elegant and best way to show a sum of products with indicies?) $\sum_{k} A_{a_{1},...,a_{i},k}*B_{k,b_{1},...,b_{j}}$ ...
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2answers
109 views

Double Binomial coefficient sum identity $S = \sum_{k = 0}^{n} (-1)^k {{4n-1-2k}\choose{2n-2k}}{{2n-1}\choose{k}}$

I have a sum of factorials that I managed to put in the following form $$S = \sum_{k = 0}^{n} (-1)^k {{4n-1-2k}\choose{2n-2k}}{{2n-1}\choose{k}}$$ where $n\in\mathbb{N}$. Mathematica can sum this ...
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1answer
53 views

How to solve this sum $\sum_{n=0}^{\infty} e^{-0.5n(n+1)}$

For my homework I need to solve the sum $$\sum_{n=0}^{\infty} e^{-0.5n(n+1)}$$ I tried to rewrite it as a geometric series because it kind of looks like one but to no success.
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0answers
35 views

Converting a Geometric Series to Sigma Notation

Have upcoming an exam on sigma notation, and one of practice questions is this; You have a geometric series $Y$ for which we have the following rule: $$Y_{t+1} = \beta_0 + \beta_1 Y_t + \beta_2 ...
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1answer
21 views

Why does the summation of these indicator variables start from i<j?

I'm currently reading through the eighth edition of A First Course in PROBABILITY, by Sheldon Ross. The section I'm reading is "Momens of the number of events that occur", and I understand everything ...
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0answers
63 views

Is there any closed expression for $\sum_{k=0}^\infty r^{k^2}$?

Let $r$ be any real number with $0 < r < 1$. Then, of course, there is a closed expression for $$\sum_{k=0}^\infty r^k = \frac{1}{1-r}$$ But does there also exist a closed expression for $$\sum_{...
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0answers
19 views

Convergence acceleration of a series by using transformation

One of the ways of accelerating the convergence of a series is by transforming into a faster series. Examples of this approach can be found in this paper. A generalization of this method leads to the ...
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2answers
43 views

Infinite Sum of Series

So I was given this question $$T_n = \sum _ {k=0}^{ n-1} \frac{n}{n^2+kn+ k^2} $$ And $$S_n = \sum _{ k=1}^n \frac{n}{n^2+kn+ k^2} $$ We were asked wether $T_n$ or$S_n$is$ \gt$or$ \lt \frac{π}{3\...
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2answers
58 views

How to interpret an “if and only if” (“iff”) statement in a summation?

I'm a programmer trying to convert the formula below into code, and I don't understand what exactly the "iff" clause on the right side of the numerator is being applied to. The fact that it ...
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1answer
18 views

Help me to understand the summation notation

I am having following expression, limits of summations are given in terms of set, can someone help me to interpret this summation. where $d_{1}$ and $d_{2}$ are distances. $$\sum_{i_1,i_2 \in \{1,2\...
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21 views

Trying to evaluate this sum : $\sum_{i=1}^ni^k(n-i)^{r-k}$

After long calculations in an exercice I end up with this sum and I wonder if you can simplify it : $$\sum_{i=1}^ni^k(n-i)^{r-k}$$ where $r$, $n$, $k$ are integers, $r\geq k$. I don't really know ...
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2answers
46 views

Exact sum of the series $\sum_{n=2}^\infty \frac{(-1)^n}{n^3-n}$

I need to find the exact sum of the following series, $\sum_{n=2}^\infty \frac{(-1)^n}{n^3-n}$. The solution goes like this: $\sum_{n=2}^\infty \frac{(-1)^n}{n^3-n}$ $= \frac12\sum_{n=2}^\infty \...
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50 views

Calculate the sum.

There is given positive decreasing sequence of real numbers $a_{1},a_{2},a_{3},...$ Calculate the sum in nice form: $$SUM=\sum_{i=1}^{N}a_{i}-2\sum_{1\le i<j\le N}a_{i}a_{j}$$ If it is impossible ...
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1answer
33 views

Deriving simple formula for summation

I am first year non-math student and I am trying to formalize my homework from digital circuits course using some basic math tools. To give some initial context: If I have $\frac{1}{5}$ frequency ...
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1answer
28 views

Summation including combination [on hold]

I am having trouble evaluating the summation: $\sum_{k=0}^{2n}(-1)^kk^n{2n \choose k}$ Can anyone lead me to a solution? Also, is there a general or easy way to approach summations that include ...
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2answers
50 views

Evaluating $\sum\frac{\sin(n)}{n^a}$

I have a function defined as: $$S(a)=\sum_{n=1}^\infty\frac{\sin(n)}{n^a}$$ My question is for what values of $a$ is this convergent, and how can I evaulate this? For starters, I know that $S(a)$ is ...
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0answers
13 views

How to bound $\sum_m e^{\mathcal{O}(k2^{m/2})-2^{3m/4}}$

I have to estimate the following sum $$ \sum_{m=0}^{\frac{2}{\log 2}\log\log\log T}e^{\mathcal{O}(k2^{m/2})-2^{3m/4}}. $$ I would like to show that this sum is $$ \ll_k 1 $$ and if possible that it ...
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1answer
33 views

are $\sum_{i=0}^{n}i$ and $\sum_{i=n}^{0}i$ equivalent?

So here's the ugly history of how I came to ask this question. I was following this proof: and got stuck at this step: $$\sum_{j=0}^{(\log_2n) - 1}\frac{1}{(\log_2n) - j} = \sum_{l = 1}^{\log_2n}\...
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0answers
23 views

Quadratic martingale bound

I know that if $a_1,a_2,\dots$ are random variables and {$\mathcal{F}_{t}$ } is a filtration such that $$\mathbb{E}[a_i \mid \mathcal{F}_{i-1}] \leq K$$ for all $i$, then for any stopping time $\...
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2answers
43 views

Floor function summation[difficult]

The question is to find the value of — $$\sum_{r=1}^{502} \Big \lfloor \frac{305r}{503}\Big \rfloor$$ The answer is pretty big, so I don't think trial and error will work here. I seriously can't ...
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1answer
27 views

Pattern in Squared Numbers and their Digit Sum

So this has been boggling my mind for some time now. On spring break I was really bored and started messing around with numbers when I noticed something. I was squaring each number (1-9) when I ...
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3answers
61 views

Sum to infinity series

Consider the following series $$\sum_{k=0}^{\infty}\frac{(k+1)(k+3)(-1)^k}{3^k}$$ I'm told to analytically find the sum to infinity and I have been given this as a clue. $$\Sigma_{k=0}^\infty x^k = ...
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82 views
+50

Evaluating sum $\sum_{m=0}^{\infty}\frac{(2-\delta_m^0)(-1)^m \lambda_0}{a(\lambda_0^2 -(\frac{m\pi}{a}))}\cos(m\pi x/a)$

How Can I evaluate the following sum$$\sum_{m=0}^{\infty}\frac{2-\delta_m^0}{a}\frac{(-1)^m \lambda_0}{\lambda_0^2 -(\frac{m\pi}{a})}\cos(\frac{m\pi x}{a})=\frac{\cos(\lambda_0 x)}{\sin(\lambda_0 a)}$$...
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0answers
11 views

Stochastic Proof Piece

As a part of the proof of recurrence criterion, (Page 22 here: http://web.math.ku.dk/noter/filer/stoknoter.pdf), it is shown that $(P^n)_{i,j} = \sum_{m=1}^n (P^{n-m})_{j,j}f_{ij}^{(m)}$. This makes ...
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2answers
45 views

How is $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}x^{i+j} = \sum_{n=0}^{\infty}\sum_{i=0}^{n}x^n$

My professor used this in class for a proof and I'm having trouble understanding it. $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}x^{i+j} = \sum_{n=0}^{\infty}\sum_{i=0}^{n}x^n$ The way he explained it, ...
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1answer
23 views

Differential Geometry - vector fields Lie bracket

Can anyone tell me why (in the last line) $i$ changes to $j$ in the first component of the sum?
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1answer
33 views

Sigma notation with equality conditional

I am used to the Sigma notation: \begin{align} \sum_{j=1}^{n} \frac{1}{j} \end{align} I have the following problem to solve: Let $H_{k}$: \begin{align} \sum\limits_{1 \le j \le k} \frac{1}{j} & ...
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1answer
19 views

Splitting / rearranging sigma sums

I am struggling to understand the concept of sigma sum rearrangement. In fact, I don't even know what to call it. That being said, if anybody can recommend sources for me to study this from or let me ...
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1answer
33 views

Values of some “natural” sums over multiindices with a given absolut value

I'd like to know if there is a nice closed expression in terms of $j$ and $k$ of the sum $$ S_{j,k}:=\sum_{(i_1,\ldots,i_k)\in \mathbb{N}^k_0:\\i_1+\cdots+i_k=j}\frac{1}{i_1!\cdots i_k!}. $$ ...
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1answer
20 views

Show that $\sum^{N-1}_{i=1} \frac{N-1}{i(N-i)}$ is approximately $2\log N$ for large $N$.

I am working on a problem relating to a Markov chain, that results in the sum: $$\sum^{N-1}_{i=1} \frac{N-1}{i(N-i)}$$ For the expected time until reaching the $N^{th}$ state from $0$. The ...
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1answer
27 views

How to prove that an intger is either of the form $2^k$ or a sum of consecutive integers. [duplicate]

I would like to show that an integer $p$ is either sum of consecutive integers or is of the form $2^k$ $k\in \Bbb N$. I know that $p$ is the sum of consecutive integers if and only if $$p =\frac{m(...
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1answer
45 views

A Tricky sum to evaluate (Haldane)

I'm trying to find a way to evaluate this sum (found by Haldane in Phys. Rev. Lett. 60, 635 (1988): $$S_{pq}=\sum_{n=1}^{N-1} z^{nJ} (1-z^{n})^{p-1}(1-z^{-n})^{q-1}$$ with $z= e^{\frac{2i\pi}{N}}$ ...
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1answer
22 views

Differential of sum of logs of probabilities

I'm trying to calculate the derivative with respect to $p1$, $p2$, ... $pk$ (each) of the following equation: $L = N_1 \log p_1 + N_2 \log p_2 + ... N_k \log p_k$ where $\Sigma_{i=1}^k p_i = 1$ i.e....
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0answers
21 views

Derivative of a sum

$$L(a,b) = {\sum_{i=1}^n (y^i - (ax^i +b)^2)^2})$$ $$\frac{dL}{da} = \sum_{i=1}^n \frac{(y^i-y^-)*(x^i-x^-))}{var(x)}$$ How can I canclulate the $\frac{dL}{da}$? $$2(y^i - (ax^i +b))^2)* \frac{d(y^i ...
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3answers
64 views

Inequality between two sums

As part of a research problem I am working on, I need to show the following inequality. Let $x=(x_1,\dots,x_K)$ with $x_i > 0$ for all $i$. Then, I wish to show that $$ \frac{1}{K^2}\sum_{i=1}^K ...
1
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1answer
63 views

To find the sum of the given infinite series

This is a question from the book 'Differential Calculus' by Joseph Edwards. Prove that if $x$ be less than unity $$\frac{1-2x}{1-x+x^2} + \frac{2x-4x^3}{1-x^2+x^4} + \frac{4x^3-8x^7}{1-x^4+x^8} \ldots ...
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0answers
22 views

Analytical solution for adding up this sequence of numbers [duplicate]

Is there an analytical solution for adding up this sequence of numbers? $$ n + \frac{n}{2} + \frac{n}{4} + \frac{n}{8} + \ldots + 1 $$ For example, let $n=32$, $$ 32 + \frac{32}{2} + \frac{32}{4} +...
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1answer
20 views

Formula for calculating the total interest payable over the life of a loan with linear redemption scheme

I am trying to calculate the lifetime interest paid on a loan with a linear redemption scheme. I know that I can enter it into a spreadsheet and take the sum of the interest, but is there a formula to ...
1
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3answers
357 views

Calculating the infinite sum $1-\frac 1 7+\frac 1 9 - \frac{1}{15} + \frac 1 {17}\mp …=\frac{1+\sqrt{2}}{8}\pi$

Prove that $$1-\dfrac 1 7+\dfrac 1 9 - \dfrac{1}{15} + \dfrac 1 {17}\mp ...=\dfrac{1+\sqrt{2}}{8}\pi$$ My attempt: I tried to break it into two series $$(1+1/9+1/17+...)-(1/7+1/15+1/23+...)$$ But I ...
0
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1answer
35 views

Finding the infinite sum using Leibniz Test

I have been given a task in my previous lecture to determine whether the infinite sum; $$\sum_{n=1}^\infty \frac{\cos (\pi n)\ln n}{n}$$ Converges or diverges. My perspective on the problem is that ...
0
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0answers
26 views

What is the partial sum formula for the variation of the following infinite series involving the Riemann zeta function?

Well known: $$\sum_{n=2}^\infty \frac{\zeta(n)-1}{n} = 1 - \gamma$$ See https://en.wikipedia.org/wiki/Riemann_zeta_function#Infinite_series So, $$\sum_{n=2}^m \frac{\zeta(2n)-1}{2n} = ?$$
2
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0answers
31 views

Joint conditional probability of geometric and poisson distributions

I'm new to the concept of joint conditionals, and I want to make sure that a move I made is valid and logic. $$X\sim Geom(0.21) \rightarrow P(X=k)=(0.79)^{k-1}\times0.21$$ $$Y|X\sim Poisson(x+1)\...
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1answer
43 views

Summing a series exactly 1 [duplicate]

How does one go about exactly summing this series $$\sum_{n=1}^{\infty}\frac{(-1)^nn^2}{3^n}$$ Stuck on this and not sure how to proceed. Appreciate any assistance!
1
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1answer
74 views

How to find the conjugate of $\sum_{n=0}^{m}n \cos(2\pi x n)$?

Supposing I want to find the conjugate of $$\tag{1}\label{eq1}\sum_{n=0}^{m}n \cos(2\pi x n)$$ If I view (1) as a Fourier series then what would be the conjugate Fourier series and how would I go ...