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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0answers
40 views

The formula for a summation for $\sum_{i=0}^n i^n,\,$ for arbitrary $n$? [duplicate]

$$\text{If }\,\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$ $$\text{and }\,\sum_{i=0}^n i^3 = \frac{[n(n+1)]^2}{4},$$ $$\text{is there a formula for }\,\sum_{i=0}^n i^n\;?$$
0
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0answers
17 views

Proving that a function has compact support

How can I argue/show in a proper way that if $f \in L^2$ has compact support and $\phi \in L^2$ with $\{T_k\phi\}_{k\in\mathbb{Z}}$ being an orthonormal system where $T_k$ is the translation operator, ...
2
votes
1answer
19 views

Summation to count number of strings over N characters?

How many different strings of five characters are there if only lower-case letters or numbers can be used in creating these strings? Here is my solution: There are 26 letters in the alphabet ...
2
votes
2answers
22 views

Taylor polynomial manipulation

Find $\sum_{k=1}^\infty\frac{(-1)^{k+1}x^k}{k}$ This is in a section in my book on Taylor polynomials/Taylor series so I assume we have to find some way to manipulate Taylor polynomials to get this. ...
2
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1answer
22 views

Semantic question about summations

This is a simple question about the syntax of summations. Is the following true: $\lambda_1\sum_\limits{j=1}^{\infty} f_1(E_j)+ ...
2
votes
2answers
46 views

How to calculate the sum $\sum_{i=0}^\infty i^nx^i$ [duplicate]

Here I have $x\in\mathbb{R}_+$ and $x < 1$. I would like to evaluate the following sum: $$\sum_{i=0}^\infty i^nx^i.$$ I know that $$\sum_{i=0}^\infty x^i=\dfrac{1}{1-x}.$$ So I started ...
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3answers
21 views

Algebraic simplification of likelihood ratio

Can someone help me understand how this: ...
1
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2answers
33 views

Standard result for $\log(x)$

$$\sum_{1\leq m\leq x/d}\frac{1}{m}=\log(\frac{x}{d})+O(1)$$ I read this result in lecture papers I was going through and can't find anything about its origin. Is there a standard summation result ...
1
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2answers
24 views

Mistake in proof of sum of divisors function $\sigma(n)$

The proof derives the correct result, but I cannot see how the first equality is correct. To begin we use the formula $\sigma(n)=\sum_{d\mid n}d$ This is the first step in the proof: $$\sum_{1\leq ...
1
vote
1answer
30 views

Expected value of Bernoulli with probability of success Gaussian distributed

I have a circle with centre $(0,0)$. I am generating Matlab code to include $N$ neurons in a neural network. The probability of including individual neurons in a network decays exponentially with ...
0
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0answers
10 views

Estimate error in approximating integral with sum

In a paper, for practical purposes the following integral is approximated as a sum, $$ \frac{1}{2\pi} \int_0^{2\pi} f(\theta) e^{-il\theta}d \theta \approx ...
2
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5answers
138 views

Estimate $\int^1_0 e^{-x^2}\, dx$

Estimate $\int^1_0 e^{-x^2}\, dx$ This is in a section on Taylor series so I would assume that is how it should be solved. I started by using the Taylor series formula for $e^x$ replacing $x$ with ...
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0answers
45 views

Hello everybody [on hold]

How does this site work? Do I just ask questions?
2
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3answers
45 views

How to express sum as triple summation

I am trying to express the following sequences as summations: $$ 1+2^2+3^2+4^4+5^4+6^4+7^4 $$ and $$ 1+(2+3)^2 + (4+5+6+7)^4 $$ as summations. I think they will likely be triple summations, so ...
0
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1answer
42 views

How do we prove $\int \frac{\ln(1+x)}{x}dx = -\sum_{k=1}^{\infty}\frac{(-x)^k}{k^2}$?

After working on the integral $\int_{0}^{1} \frac{\ln(1+x)}{x}dx$ for a couple of hours, I became convinced its antiderivative was not elementary. So I looked it up on Wolfram Alpha, and it found that ...
1
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1answer
51 views

Prove summations are equal

Prove that: $$\sum_{r=1}^{p^n} \frac{p^n}{gcd(p^n,r)} = \sum_{k=0}^{2n} (-1)^k p^{2n-k} = p^{2n} - p^ {2n-1} + p^{2n-2} - ... + p^{2n-2n}$$ I'm not exactly sure how to do this unless I can say: ...
1
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1answer
29 views

Closed Form Summation Example

$$ \sum_{i=1}^n (ai +b) $$ Let $n \geq 1$ be an integer, and let $a,b > 0$ be positive real numbers. Find a closed form for the following expression. In other words you are to eliminate the ...
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1answer
17 views

Summation to Closed Form conversion

I am struggling to understand basics as it related to forming a closed form expression from a summation. I understand the goal at hand, but do not understand the process for which to follow in order ...
2
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2answers
32 views

binomial sum binomial (a + k , a)

Anyone know a way to compute such a sum : $$S = \sum_{k=0}^{n}\binom{a+k}{a} $$ I encountered this sum in a problem in which $a=7, n=7$. In this case the sum can be computed by hand, but I was ...
0
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1answer
33 views

How to evaluate a squared sum?

I need (the name of) the formula to evaluate $(x_1 + x_2 + ... + x_n ) ^2$ . I know the question is not very interesting, but I am stuck and WolframAlpha also doesn't get my input. Thanks in advance
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2answers
26 views

Show $\sum_{1\leq n\leq x}\sum_{d\mid n}f(d)=\sum_{1\leq d\leq x}\sum_{1\leq m\leq x/d}f(d)$

I have been trying to get my head around this step in a proof, but havn't been able to, Question: Show $$\sum_{1\leq n\leq x}\sum_{d\mid n}f(d)=\sum_{1\leq d\leq x}\sum_{1\leq m\leq x/d}f(d)$$ ...
1
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1answer
26 views

Proof By Induction - $n^2 = \sum_{i=1} ^{n} (2i-1)$ for all $n\geq 1$ [duplicate]

Using Proof By Induction I am trying to prove the following: $n^2 = \sum_{i=1} ^{n} (2i-1) $ for all $n\geq 1$ Here is my solutions so Far: Base Case: $n=1, LHS: 2(1)-1 = 1, RHS = 1^2 = 1, True$ ...
2
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1answer
26 views

Changing order of summation including a min in the summation

Lets say I have the following expression: $$ h(x) = \sum_{k=1}^n \sum_{v=1}^{\min\{k,j\}} \frac{(-1)^{n-k}k!}{(k-v)!} {n \brack k}f(x)^{k-v} B_{n,v}^f(x) $$ Now my goal is to have the $v$ ...
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0answers
64 views

Find the upper and lower sum of an integral with a floor

I'm having some trouble and looking for some help with a problem i'm trying to solve. Without the floor function it would be easy but the floor has made it a bit trickier: Find the upper and lower ...
0
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1answer
20 views

Changing the limits of a summation

Sorry about bad englsih Guys, i have this: n n-1 ( Σ 3k²-k) + ( Σ 2k-3k²) k=1 k=0 So, the limits of the first one are: ...
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2answers
49 views

Proving that a summation is multiplicative

I have been give a project for number theory: For $m>0$ , let $f(m) = \sum_{r=1}^m \frac{m}{\gcd(m,r)}$ . Evaluate $f(m)$ in terms of the prime factorization of $m$. So far, I have found a formula ...
3
votes
1answer
68 views

How prove that $\frac{1}{\sin^2\frac{\pi}{2n}}+\frac{1}{\sin^2\frac{2\pi}{2n}}+\cdots+\frac{1}{\sin^2\frac{(n-1)\pi}{2n}} =\frac{2}{3}(n-1)(n+1)$ [duplicate]

How prove that sum $$\frac{1}{\sin^2\frac{\pi}{2n}}+\frac{1}{\sin^2\frac{2\pi}{2n}}+\cdots+\frac{1}{\sin^2\frac{(n-1)\pi}{2n}} =\frac{2}{3}(n-1)(n+1)$$
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4answers
96 views

Solve the recurrence of the alternating sum $R_n=R_{n-1}+(-1)^{n}(n+1)^{2}$

I have been trying to solve this recurrence for a few hours, but I haven't been able to find the solution yet: $R_0=1$ $R_n=R_{n-1}+(-1)^{n}*(n+1)^{2}$. I have been trying to substitute ...
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3answers
43 views

How do I find a solution to this this finite series? $ \frac{1}{n^4} \sum_{i=1}^{n} \left({i^3}\right) $

How do I find a solution to this this finite series? Any help would be greatly appreciated. $$ \frac{1}{n^4} \sum_{i=1}^{n} \left({i^3}\right) $$
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0answers
57 views

How to simplify $\sum_{r=1}^{y} \binom{x-1}{r}\binom{y-1}{r}$? [on hold]

To find sum of the product of two combination terms $$\sum_{r=1}^{y-1} \binom{x-1}{r}\binom{y-1}{r}$$
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1answer
42 views

to simplify the following combinatorial terms [on hold]

To simplify the following summation involving product of combinations $$\sum_{r=1}^{y}\left(\begin{array}{c} x-1 \\ r \end{array}\right) \left(\begin{array}{c} y-1 \\ r-1 ...
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2answers
58 views

Proof By Induction [on hold]

I am trying to prove the Following, However, I dont understand what to do at the Inductive Step: Any Help would be appreciated!
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0answers
59 views
+50

Find Functions That Can Be Inverted from Their Sums

I have the following situation:$$ f_1(x_1) + f_1(x_2) + f_1(x_3) + \cdots + f_1(x_n) = c_1\\ f_2(x_1) + f_2(x_2) + f_2(x_3) + \cdots + f_2(x_n) = c_2\\ \vdots\\ f_n(x_1) + f_n(x_2) + f_n(x_3) + ...
1
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1answer
34 views

Proving the equality of sequences' limit [duplicate]

Prove that if $a_n$ is convergent, then $M_n:=\frac{1}{n}\sum_{1}^{n} a_n$ satisfies $\lim M_n=\lim a_n$. (sorry for English)
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1answer
25 views

Summation. Combining different set of indices.

I am reading the second chapter of Concrete Mathematics book and I cant get my head aroud a simple concept: it is stated there that $$ \sum _{k \in K} a_k + \sum _{k \in K'}a_k = \sum _{k \in K \cap ...
2
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1answer
152 views

How can I prove this combinatorial identity?

Let $n,m$ be non-negative integers. How can one prove the following identity? $$\sum_{j=0}^n j\binom{2n}{n+j}\binom{m+j-1}{2m-1}=m\cdot4^{n-m}\cdot\binom{n}{m}$$
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0answers
28 views

Dealing with constants in summation notation

I haven't worked with summation notation in a while, and am unsure how to approach the following: $\sum_1^n [-\frac 12 * \frac{(x_i - \alpha)^2}{\alpha}]$ where $\alpha \in R$ What would be the ...
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2answers
33 views

Calculating a power series [closed]

I was wondering if anyone knows how to calculate: $\sum_\limits{t=-\infty}^{\infty}$ $a^{t} e^{-itb}$, for constants a,b and $-\pi < b < \pi$ Can we take the t=0 term out to reduce it ...
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0answers
44 views

Infinity Series [closed]

Good afternoon. I'm brazilian, then sorry by my bad english. I have a problem with one question about Infinite Series. I searched for anyone method could help me. I have all constants values (w, y, ...
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0answers
24 views

Infinite series

Good afternoon. I'm brazilian, then sorry by my bad english. I have a problem with one question about Infinite Series. I searched for anyone method could help me. I have all constants values (w, y, ...
2
votes
0answers
31 views

Sum of Floor of Square Root: $S = \sum_{k=1}^{n} \lfloor \sqrt{k}\rfloor$

$$ S = \sum_{k=1}^{n} \lfloor \sqrt{k}\rfloor. $$ Hello, I´m trying to solve this summation. I was able to get $$ a_n = 2a_{(n-1)} - a_{(n-2)} $$ for non perfect square numbers and $$ a_n = ...
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0answers
24 views

inclusion–exclusion principle formula in probability for general case

For the system reliability analysis of complex systems, I'll use the formula below after a point in a custom computer code. I know that for the n=2 case, formula result is $P_1+P_2-P_1.P_2$ ($P$ : ...
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votes
3answers
45 views

Summation of $3^k$ from $2$ to $72$ [closed]

i'm currently stuck on the following question and am not supposed to be using a summation calculator to find the answer: $$ \sum _{k=2}^{72}\left(3^k\right) $$ Please could somebody explain to me ...
1
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1answer
20 views

Sum identity involving sin

How one can prove that $$\sum_{k=1}^n(-1)^k\sin(2k\theta)=\cos(n\pi/2+\theta+n\theta)\sec\theta\sin(n\pi/2+n\theta)?$$ It looks difficult as there is sum on the other side and product of trigonometric ...
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0answers
43 views

Summation of trigonometric series

The first part of the question requires me to show that the sum of $$cos(2n - 1)x = \frac{sin(2Nx)}{2sin(x)}$$ from $n = 1$ to $N$. This I have done by considering the real part of the geometric ...
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3answers
87 views

approximation of a sum $\sum ^{\sqrt{n} }_{k=5}\frac{\log\log(k)}{k\log(k)} $ [closed]

What is the approximation of this sum $$\sum^{\sqrt{n}}_{k=5}\frac{\log\log(k)}{k\log(k)}$$ Should I proceed by an integral ? How can I calculate its lower and upper bound?
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0answers
30 views

Asymptotic Formula for Sum

I am trying to find an asymptotic formula for the sum of the following: $\sum _{x=1}^{\infty } x \left(\left(1-\frac{\Gamma (x,\lambda )}{\Gamma (x)}\right)^n-\left(1-\frac{\Gamma (x+1,\lambda ...
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0answers
22 views

Combinatorial identity binomial coefficients [duplicate]

How to prove that $$ \binom{m}{p} = \sum_{j=0}^q \binom{q}{j}\binom{m-q}{p-j}\;?$$
3
votes
2answers
20 views

notation for Sumation of Sumation for only for odd iterations

I need to write a summation in summation whether the inner summation should iterate from one through all odd numbers to the teration of the outer summation which goes from 1 to $\infty$... Something ...
0
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0answers
26 views

Symbol for exponentiation of a sequence? (Equivalent to SIGMA for summation and PI for product)

I have a student asking whether there is a symbol for exponentiation of a sequence? So there's SIGMA for summation of a sequence, PI for multiplication of a sequence and perhaps something else for ...