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

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2
votes
2answers
39 views

Prove by induction that $\sum_{k=1}^nk^p < (n+1)^{p+1}/(p+1), \quad n,p \in \mathbb{N}$

For $n=1$, we have at the left side $1^p$, and at the right side: $$ \frac{2^{p+1}}{p+1}\mathrm{~which~is } >1$$ so it holds for $n=1$, but how can we prove that $$ \sum_{k=1}^{n+1}k^p<\frac{(n+...
0
votes
0answers
9 views

How to compute this sum over values of the derivative of the sinc function?

If $g(t)=\frac{sin(\pi t)/T}{\pi t/T}$ and $g'(t) = \frac{\partial}{\partial t}g(t)$, then how to compute this sum? $$SUM = \sum_i a_i \sum_m h_m g'(kT - iT - \tau_k -mT),$$ where $\{a_i\} \in \{\pm ...
0
votes
2answers
28 views

Find base of exp function within the range of summation

I got a sequence where the relation between elements of the sequence is given by: \begin{align} y_1 &= b \\ y_{i+1} &= 2 y_i + b \quad (i \in \mathbb{N}) \end{align} where $b$ is called base, $...
0
votes
1answer
40 views

The limit of $\lim_{\Delta{x}\to0}\sum{_0^\infty}(2x\Delta{x})$

I was trying to derive integrals of some elementary function through summation notation. It was-- $\lim_{\Delta{x}\to0}\sum{_0^\infty}(2x\Delta{x})$
3
votes
2answers
60 views

Hard summation involving binomial and quadratic

What is $$\sum \frac{2r^2-98r+1}{(100-r)({100\choose r})}$$ Where $r\in [1,99]$I have reduced it to $$\frac{(2r^2-98r+1)}{(100){99\choose r}}$$ what to do further? Partial fractions don't seem to ...
2
votes
2answers
140 views

$ 1^k+2^k+3^k+…+(p-1)^k $ always a multiple of $p$?

I would appreciate if somebody could help me with the following problem: Q: For any prime number $p(p\geq 3), k=1,2,3,...,p-2$, why is $$ 1^k+2^k+3^k+...+(p-1)^k $$ always a multiple of $p$ ?
4
votes
4answers
308 views

Sum of sum of binomial coefficients

I know there is no simple way to solve the sum: $$\sum_{k=0}^{j}{{n}\choose{k}}$$ But what about the equation: $$\sum_{j=1}^{n}{\sum_{k=0}^{j}{{n}\choose{k}}}$$ Are there any simplifications or ...
1
vote
1answer
34 views

How can I write a set of equations in summation form?

I have a system of equations as follows: \begin{align} & A_1^{11} + A_1^{12} + A_1^{13} + \cdots + A_1^{1n}=X \\[8pt] & A_1^{21} + A_1^{22} + A_1^{23} + \cdots+ A_1^{2n}=X \\[8pt] & \...
12
votes
3answers
469 views

How can we show that $ \sum_{n=0}^{\infty}\frac{2^nn[n(\pi^3+1)+\pi^2](n^2+n-1)}{(2n+1)(2n+3){2n \choose n}}=1+\pi+\pi^2+\pi^3+\pi^4 ?$

We proposed this sum, but we are lacking in knowledge of this area of maths and we would ask if any of the authors would be willing to show us step by step how to go about proving this sum. $$ \sum_{n=...
3
votes
2answers
69 views

If $\sum_{n=1}^{\infty}\frac{1}{{n}^2} = \frac{{\pi}^2}{6}$ then $\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}$ is equal to:

If $\sum_{n=1}^{\infty}\frac{1}{{n}^2} = \frac{{\pi}^2}{6}$ then $\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}$ is equal to: I do not know what to try to find the solution. A hint along with the explanation ...
7
votes
3answers
66 views

Is the sum of reciprocals of all products from $2$ to $n-1$ always $0.5n-1$?

I was looking up riddles for my math classes to work on for the end of the year and found the following riddle. http://mathriddles.williams.edu/?p=129 I followed the advice and started working with ...
0
votes
2answers
60 views

What's wrong with my Taylor -Maclaurin- Series? $e^{x^2+x}$

Here's what I have: We know: $$e^x = 1 + x + \frac{1}{2!}x^2+\frac{1}{3!}x^3 +\frac{1}{4!}x^4$$ Now I can calculate the Taylor Series for $e^{x^2+x}$: $$1+u+u^2+\frac{1}{2!}(x^2+x)^2+\frac{1}{3!}(...
1
vote
2answers
64 views

Is there any closed form expression for the following sum $\sum_{t=1}^T \frac{2^t}{t} = ?$

Is the result upper bounded by $E_i(\ln(2)T)$ ? Edit: where $$E_i(y) = \int_{-\infty}^y \frac{\exp(z)}{z} \mathbb{dz}$$
1
vote
3answers
55 views

Why is $\sum_{i=0}^{+\infty} a^{i}i=\frac{a}{(1-a)^{2}}$?

I saw this series in some mathematical proofs but I couldn't find why $\sum_{i=0}^{+\infty} a^{i}i=\frac{a}{(1-a)^{2}}$
2
votes
1answer
22 views

Deriving the identity: $\hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$

For some reason I am having an extremely hard time finding out how the following expression is derived $$ \hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} $$ Is ...
6
votes
3answers
138 views

My formula for sum of consecutive squares series?

I stumbled upon a specific series, who's Sum of squares of consecutive integers equals the sum of squares of the continuation of that consecutive integers. For exmaple, this first number in the ...
1
vote
1answer
18 views

Accurate summation of mixed-sign floating-point values

Due to the finite representation, floating-point addition loses significant bits. This is particularly noticeable when there is catastrophic cancellation, such that all the significant bits can ...
0
votes
1answer
98 views

Is there any summation method that assigns $ \sum_{n=1}^\infty \frac{1}{n} =-\frac{\pi}{2}$

I don't know too much about alternate summation methods, but am interesting to know if any give the sum of the harmonic series to be $$-\frac{\pi}{2}$$
2
votes
1answer
43 views

Summing Over Uncountable Index Sets

In answering the question Why do we classify infinities in so many symbols and ideas?, William's answer asserted that summing over an uncountable index set necessarily results in an infinite sum. I am ...
0
votes
1answer
66 views

Solving a series of equations

I'm writing a piece of code to translate some data, and I keep banging my head against the wall with a one part of the transformation. I'm not as good at math as I ought to be :) Say we have a ...
2
votes
1answer
37 views

Summation with arithmetic series

I have doubts how to solve summation if on the top there is something else than $n$ more specificially: $$\sum\limits_{i=1}^{n+1} i=\frac{(n+1)(n+2)}{2}$$ Is my solution correct? I just replace $n+1$...
4
votes
1answer
399 views

Sum over fourth power of the sine

I am considering the sum $$ A_m = \sum_{j=0}^m \sin^4\left(\frac{j}{m}\cdot\frac{\pi}{2}\right). $$ I think that for $m>1$ it holds $$ A_m = \frac{3m+4}{8}, $$ but I can't really get to it.
0
votes
1answer
29 views

Matrix Notation! (Linear Algebra)

Suppose that we have a NxM matrix, where N=rows and M=columns. How could I write nicely a ...
2
votes
2answers
46 views

Evaluate the following trignometric sum

I am interested in the following sum $$\sum_{\text{even } n=-\infty}^{\infty}\left(-\cos^2x\delta_{n,0}+\cos x\left(\frac{1-\cos x}{\sin x}\right)^{|n|}\right).$$ Wolfram alpha returns answer $\sin^...
0
votes
1answer
35 views

Infinite sum of Hermite polynomials with same order, but different argument

I am looking for any possible simplification of the following sum for positive reals $\alpha,\beta$ and positive integer $n$: $$ \sum_{t=-\infty}^{\infty}e^{-\beta(t+\alpha)^{2}}H_{n}(t+\alpha) $$ I'...
0
votes
1answer
21 views

Can somone explain this specific Summary Notation to me?

Been re-teaching myself all the math I slept through in high school. Really enjoying the challenge, but I hit a wall last night. The workbook I have has the following notation: $$\prod_{i=0}^4 7(i + ...
0
votes
2answers
42 views

How to solve this series :$\sum_{k=0}^{\frac{n}{2}}n-k$

I tried to solve this series as follows ; $\sum_{k=0}^{\frac{n}{2}}n-k$ : $ =(\frac{n}{2}+n)+(\frac{n}{2}+1+n-1)+(\frac{n}{2}+2+n-2)+...+(\frac{n}{2}+k+n-k) = \frac{n}{2}(\frac{n}{2}+n)=\frac{n^2}{4}+...
0
votes
1answer
18 views

Formulae for $\sum_{k=0}^{\left\lfloor n/2\right\rfloor }\left(\begin{array}{c} n\\ 2k \end{array}\right)(-1)^{k}a^{n-2k}b^{2k}$?

Given a complex number $z=a+bi$, its $n$th power can be written in closed form as $$(a+bi)^n=\sum_{k=0}^{\left\lfloor n/2\right\rfloor }\left(\begin{array}{c} n\\ 2k \end{array}\right)(-1)^{k}a^{n-2k}...
0
votes
1answer
26 views

implementing double $\sum_{m=1}^{p}\sum_{n=1}^{\infty }$ sum in MATLAB

I want to calculated following double sum: $$\sum_{m=1}^{p}\sum_{n=1}^{\infty }(-1)^{n-1}\Delta H.{erfc\frac{2(n-1)u+u\gamma}{2\gamma \sqrt{p-m}}+erfc\frac{2nu-u\gamma}{2\gamma \sqrt{p-m}} }$$ I have ...
0
votes
1answer
107 views

Proving the closed form of a generating function of the sum of n lucas numbers is equal to the n+2th lucas number

1760887     I've been working on this homework problem for a while now and can't seem to solve it. Let $L_n = L_{n-1} + L_{n-2}$ for $n\ge 2$ where $L_0 = 2$ and $L_1 = 1$ $M_n = 1 + \sum_{i=0}^n{...
3
votes
0answers
39 views

Find the sum of $\binom{2007}{0}+\binom{2007}{4}+…+\binom{2007}{2004}$ [duplicate]

Find the sum of $$S=\binom{2007}{0}+\binom{2007}{4}+\binom{2007}{8}+...+\binom{2007}{2004}$$ My work so far: $$(1+1)^n=2^n=\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n}$$ $$(1-1)^n=0=\...
1
vote
1answer
36 views

Identity for the sum of products of Sinc functions

The Sinc function is defined as follows: $$\mathrm{sinc}(t) = \begin{cases} \frac{\sin(\pi t)}{ \pi t} & \mathrm{if} \quad t \neq 0, \\ 1 & \mathrm{otherwise.} \end{cases}$$ I want to show the ...
17
votes
0answers
316 views

Simplify the sum $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$? - a sum shows all primes $\le N^2$

I was looking for a closed form but it seemed too difficult. Now I'm seeking help to simplify this sum. The 50 bounty points or more will be awarded for any meaningful simplification of this sum. I ...
1
vote
2answers
44 views

Alternating series of compositions of triangular numbers

I'm modeling a process which involves a subset $S$ of a large number $n_A$ of objects - call them balls. Each time I add a ball to $S$, it may dislodge another ball with probability proportional to ...
5
votes
1answer
80 views

Combinatorics problem that deals with trigonometric functions

If $m$ and $p$ are positive integers and $m \geq p$, then show that $${m \choose 0}+{m \choose p}+{m \choose 2p}+{m \choose 3p}+\cdots$$ has value $${2^m \over p}\left(1+\sum_{k=1}^{\left \lfloor {...
1
vote
2answers
35 views

Is there a simple formula for calculating $\sum_r\binom{n}{r}$, where $1 \leq r \leq n$?

I am aware that the formula for combinations is: $$\binom{n}{r}=\frac{n!}{(n - r)! \:r!}$$ This gives me the numbers I want for combinations of a certain length. However, in my case, I need the ...
0
votes
1answer
20 views

Laplace pairs - proof of summation transform

I am studying this question for my finals revision and I'm lost on how to start it, can anyone suggest something? Probably pretty simple but I've hit a dead end. Here's the question: If $F_i(t)$ ...
2
votes
4answers
108 views

Direct Proof for sum of $n$ integers equation?

I am trying to prove by direct proof that $$3+5+7+\ldots+(2n+1)=n(n+2)$$ for all natural numbers $n$. I figured out how to do it by induction, but I know it can be done directly and I can't ...
0
votes
1answer
31 views

Can someone draw a plot for this function?

Can someone draw a plot for this function? $ f(x) = \begin{cases} \sum_{i=2}^{x}\left(\frac{\prod_{k=1}^{i-1}\left(2k-1\right)\,\cdot\,-\left(-\frac{1}{2}\right)^{i}}{i!}\right) + \frac{3}{2} & x ...
1
vote
1answer
58 views

$a_n = b_n -b_{n-1}$ Prove that $\sum_{n=1}^{\infty} a_n$ converges iff $\lim_{n \to \infty} b_n$ exists

Let $\{b_n\}$ be a sequence Let $a_n = b_n - b_{n-1}$. Prove that $\sum\limits_{n=1}^{\infty} a_n$ converges iff $\lim_{n \to \infty} b_n$ exists. I am extremely stuck on this homework problem and ...
0
votes
0answers
15 views

expression for a rational number

Let $x=\sum_{k=1}^n-b_kg^{-k}$ where $g\in \Bbb N$ and $b_k\in\{0,\cdots,g-1\}$. I want to write $x$ in the form $x=m+\sum_{k=1}^nc_kg^{-k}$ with $c_k\in\{0,\cdots, g-1\}$ and $m\in \Bbb Z$. Doing ...
1
vote
0answers
41 views

Can you get the average order of $ \left( 1+|\mu(n)| \right)^{M(n)} $, where $\mu(n)$ and $M(n)$ are the Möbius and Mertens functions, respectively

When yesterday I was interested in do a little study about the arithmetic function $$f(n)=\left( 1+|\mu(n)| \right)^{M(n)},$$ defined for integers $n\geq 1$, which $\mu(n)$ is the Möbius function and $...
1
vote
2answers
33 views

Find general formula for $\sum _{i=1}^{n} \frac {(-1)^i i}{(2i-1)(2i+1)}$

I was able to find formulas for simpler expressions but I can't find the general formula for this one: $\sum _{i=1}^{n} \frac {(-1)^i i}{(2i-1)(2i+1)}$ I don't see any particular trend that would ...
0
votes
2answers
23 views

General formula for a summation

I can't find the general formula for the following sum. $q \in \Bbb R, q \ne 1$ $\sum _{i=0}^{n} q^{2i}$ Any hints?
1
vote
1answer
37 views

How to expand the summation term with power?

How to expand the following: $$ \left( \sum^{M}_{m=0} \frac{x^{m}}{m!} \right)^{K} $$ where $M$ and $K$ are positive integers.
0
votes
1answer
75 views

$\sum_{n=1}^{\infty} ne^{-2n}$ estimate to 4 decimal places

I am supposed to estimate the sum correct to 4 decimal places and assume it converges. I know that I am supposed to plug in numbers for $n$ (Instructor says that solving for $n$ is impossible) however ...
1
vote
1answer
15 views

Solution check: summation inequality proof by induction

I'm not sure if what I've done works or if it's proof enough. (I need to prove that the inequality is true $\forall n \in \mathbb{N}$). $\sum_{i=n}^{2n} \frac{i}{2^i} \leq n$ $P(1)$ works. I assume ...
1
vote
1answer
36 views

Summation of $\int\frac{1}{1+x}dx$ in a range of 1 to infinity. [closed]

Let $0 < \alpha < \beta < 1$. Then $$\sum_{k=1}^\infty \int_{1/(k+\beta)}^{1/(k+\alpha)} \frac{1}{1+x}dx$$ is equal to $$ \begin{align} &(A)\ln \frac{\beta}{\alpha}\qquad\qquad (B)\ln\...
0
votes
1answer
47 views

If $\ (1+3x+x^2)^{10}=\sum_{r=0}^{20}a_r x^r\ $ then…

If $$\ (1+3x+x^2)^{10}=\sum_{r=0}^{20}a_r x^r\ $$ Then then what is the least number except 1 which divides the following:$$\ \sum_{r=0}^{20}(3r+1)a_r\ $$ EDIT: i have put x=1 then it is something ...
2
votes
3answers
67 views

$\sum_{k=0}^{2016}(1+ \omega^k)^{2017}\ $

Let $\omega \ $be a root of the polynomial $\ x^{2016} +x^{2015}+x^{2014}+...+x+1=0 \ $. Then find the value of the following sum: $$\sum_{k=0}^{2016}(1+ \omega^k)^{2017}\ $$ Well I have simplified ...