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

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1
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3answers
61 views

Prove $\sum_{k=0}^{n}{n \choose k}(-1)^k \frac{1}{k+1} = \frac{1}{n+1} $

I am going through a proof of the combinatorial result $$\sum_{k=0}^{n}{n \choose k}(-1)^k \frac{1}{k+1} = \frac{1}{n+1} $$, and have found something I don't understand. The proof is as follows: $$\...
2
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1answer
30 views

Cascading Summation (2) $\sum_{i_1\le i_2\le i_3\le \cdots \le i_m}^n \left[\prod_{r=1}^m (i_r+2r-2)\right]/(2m-1)!!$

Evaluate the following summation: $$\large\sum_{i_1\le i_2\le i_3\le \cdots \le i_m}^n \left[\prod_{r=1}^m (i_r+2r-2)\right]\bigg /(2m-1)!!\\ =\large\sum_{i_1=1}^n\sum_{i_2=i_1}^n\sum_{i_3=i_2}^n\...
1
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0answers
14 views

Rearranging summation terms including a complex exponential expression

I'm reading a paper on signal processing and having a hard time wrapping my head around a step the author takes. The signal of interest is defined as $r_k = e^{j(2\pi\Delta f k T_s + \theta)} + v_k$ ...
6
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1answer
60 views

Combinatorial argument for $1+\sum_{r=1}^{r=n} r\cdot r! = (n+1)!$ [duplicate]

Combinatorial argument for $$1+\sum\limits_{r=1}^{r=n} \ r\cdot r! = (n+1)!$$ The algebraic proof is easy as $r=(r+1)-1$.
6
votes
1answer
46 views

Sum of the reciprocal of the prime-position primes.

The primes are $2, 3, 5, 7, 11, 13...$ The sum of the reciprocals of the primes diverges, proven by Euler: $$\sum_{n=1}^\infty{\frac{1}{p_n}}=\infty$$ Here, $p_n$ is the $n$-th prime. I'm asked to ...
0
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0answers
24 views

Sum notation $\sum_{\sigma\in\{\pm 1\}^n}$?

I would like to know what the following sum notation means: $$\sum_{\sigma\in\{\pm 1\}^n}\left(\prod_{1\leq i\leq n}F(x_i^{\sigma_i})\right)$$ where $n$ is a positive integer, $x_i$ are some ...
0
votes
1answer
31 views

Can you take the limit as $x \to \infty$ of an expression such as $\sum_{n \in \mathbb{Z}} \ln(|x - n|)$? [closed]

Consider the function $f(x) = \sum_{n \in \mathbb{Z}} \ln(|x - n|)$ I'm not really concerned with its convergence properties, what I am concerned with is if it is possible to take the limit as $x \...
0
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2answers
58 views

Evaluate the following sums using generating functions

I have two series that I'm supposed to evaluate using generating functions. (a) $0+1+2+3+4+ ...+ n$ (b) $0 + 3 + 12+...+3n^2$ I know how to evaluate (a) using walks in Pascal's triangle: the answer ...
2
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2answers
94 views

Cascading Summation $\sum_{i=1}^n\sum_{j=i}^n\sum_{k=j}^n \frac {i(j+2)(k+4)}{15} $

Evaluate $$\sum_{i=1}^n\sum_{j=i}^n\sum_{k=j}^n \frac {i(j+2)(k+4)}{15} $$ Background Many basic summation questions on MSE relate to a single index - it might be interesting to devise a ...
0
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1answer
29 views

Infinite sum over Gamma functions?

I am having quite a bit of trouble understanding this sum. Can someone explain to me exactly how to this from 1 to 3,very easily way? Question its from this webpage Thanks.
4
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2answers
122 views

Infinite Sum without using $\sin\pi$

What's a purely algebraic way to prove that $\pi-\frac{\pi^3}{3!}+\frac{\pi^5}{5!}-\dots=0$? I'm sure that the first step is to write $\pi=4-\frac43+\frac45-\dots$, but I haven't been bold enough to ...
1
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2answers
23 views

Double Sigma with nested index, $i$

I am trying to solve this double sigma but my answer doesn't seem right. $$\sum\limits_{i=1}^n \sum\limits_{j=i}^{n^2}1=-i\sum\limits_{i=1}^n n^2 = -in^3$$
8
votes
1answer
81 views

Trigonometric proof stuck with induction step

I am trying to prove: $$\sum_{s=0}^{\infty}\frac{1}{(sn)!}=\frac{1}{n}\sum_{r=0}^{n-1}\exp\left(\cos\left(\frac{2r\pi}{n}\right)\right)\cos\left(\sin\left(\frac{2r\pi}{n}\right)\right)$$ We know that ...
0
votes
1answer
21 views

Number of lattice points in triangle formed by x-axis, y-axis and given line

Given a line $ax+by=c$ where $a,b,c$ are positive integers. Is there any formula to find the number of points inside the triangle formed by this line, $x$-axis and $y$-axis? Points on the boundary ...
0
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1answer
21 views

Turning points of a weighted cosine basis sum

I'm doing some work with a cosine basis, where in the interval $[0, \pi]$ some function $f(x)$ is given by $$ f(x) = \sum_{n = 0}^{M} a_n \cos{\left( nx \right)} $$ For a given set of coefficients $\...
2
votes
0answers
38 views

Pull constant out of a summation of fractions

General problem $$ \sum_{i=1}^n \frac{a_i + x}{b_i + x} = 0 $$ Is it possible for solve for $x$? Some context I've hit a road block in my derivation... At this point, I need to pull the model ...
1
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2answers
43 views

Changing summation in a power series

I'm doing a question in my power series unit that involves adding summations together, I just started this unit so I'm not totally clear on how changing summation works, from what I understand you ...
1
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3answers
67 views

Find explicit formula for summation

I have this summation: $\displaystyle\sum_{i=1}^{\log_2 n} 2^{i}$, any suggestion of how get an explicit formula?
-1
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1answer
109 views

Summation notational convention

Please correct improper notation/terminology $$\sum_{k=0}^{n-1} ar^k$$. $$\sum_{k=1}^{n} ar^{k-1}$$ As far as I can tell these both represent the same thing. It's the partial sum {$S_n$} where the ...
2
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1answer
22 views

Solving differential equations problem with Unit-step Function and Delta Function.

Suppose we have a monic differential equation defined by $$x'' + x = \sum_{n=0}^\infty \delta(t - 2n\pi)$$ When we take the Laplace transform of this differential equation, we get $$s^2X(s) + X(s) = \...
2
votes
3answers
82 views

Sum of a series $\frac {1}{n^2 - m^2}$ m and n odd, $m \ne n$

I was working on a physics problem, where I encountered the following summation problem: $$ \sum_{m = 1}^\infty \frac{1}{n^2 - m^2}$$ where m doesn't equal n, and both are odd. n is a fixed constant ...
0
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0answers
32 views

How to induct for double summation?

I have no idea on how to approach this? $$ \sum_{i=1}^{n}\sum_{j=1}^{m}a_i + a_j = \sum_{i=1}^{n} a_i + \sum_{i=1}^{m} a_j (it \space may \space be \space wrong \space it \space is \space just ...
4
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3answers
123 views

Upper bound for $\sum_{n=1}^xn^{k-1}$

From some Calculus and guess-work, I found that $$k\sum_{n=1}^xn^{k-1}<(x+\frac12)^k\tag1$$ In fact, I found that it was very, very, close. And, from even more guesswork, $$\lfloor(x+\frac12)^k\...
0
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1answer
31 views

Nice formula for a sum product

So suppose I have an ordered set of numbers: $(a_1, a_2, ..., a_n)$ and I want to express the following sum/product in an elegant manner: $ a_1 + a_1 a_2 + a_1 a_2 a_3 + ... + a_1 a_2 ... a_n $ I ...
0
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0answers
26 views

Resolving Zeros in Product of items in list.

Given the formula: $\sqrt [ 1/N ]{ \prod _{ n=1 }^{ N }{ { P }_{ n } } } $ where ${ P }_{ n }$ is a list of real numbers, e.g. [0.4, 0.3, 0.2, 0.1] And the ...
4
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3answers
192 views

Find formula for $\frac{1}{\sqrt 1}+ \frac{1}{\sqrt 2}+\cdots+\frac{1}{\sqrt n}$

I have the series: $$\frac{1}{\sqrt 1}+ \frac{1}{\sqrt 2}+\cdots+\frac{1}{\sqrt n}$$ I find hard to generalize into one formula, any explanation would be helpful.
1
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1answer
57 views

Sums involving floor function

I am looking for a direct formula for this sum $$\sum_{k=0}^n \lfloor{\sqrt{n+k}}\rfloor\lfloor{\sqrt{k}}\rfloor$$ Or a method to efficiently compute the sum for large n
4
votes
4answers
195 views

How do I show that $\sum_{k = 0}^n \binom nk^2 = \binom {2n}n$? [duplicate]

$$\sum_{k = 0}^n \binom nk^2 = \binom {2n}n$$ I know how to "prove" it by interpretation (using the definition of binomial coefficients), but how do I actually prove it?
4
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2answers
39 views

How to derive $\sum j^2$ from telescoping property

The book Real Analysis via Sequences and Series has a method of proving that $$\sum_{j=1}^n j = \frac{n(n+1)}{2}$$ that I've never seen before. The way they do it is by starting with $\sum (2j+1)$, ...
1
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1answer
45 views

Finding an upperbound for $\sum_{i=2}^{n}\bigg(\prod_{k=2}^{i}\dfrac{p_k-2}{p_k}\bigg)$

I was wondering whether there exists a known upperbound for: $$f(n)=\sum_{i=2}^{n}\bigg(\prod_{k=2}^{i}\dfrac{p_k-2}{p_k}\bigg)$$ For example: $$f(4)=\dfrac{1}{3}+\dfrac{1\cdot3}{3\cdot5}+\dfrac{1\...
4
votes
3answers
77 views

Last Digit of $x^0 + x^1 + x^2 + \cdots + x^{p-1} + x^p$

Given $x$ and $p$. Find the last digit of $x^0 + x^1 + x^2 + \cdots + x^{p-1} + x^p$ I need a general formula. I can find that the sum is equal to $\dfrac{x^{p+1}-1}{x-1}$ But how to find the ...
1
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0answers
36 views

How to find function $F$ such that $F''(x)=\cos{x^2}$, $F'(0)=3$ and $F(0)=4$?

Here we want $F\in \Bbb{R}$. We use Taylor Series. I get $F''(x)=\cos{x^2}=\sum^\infty_{k=0}\frac{(-1)^k}{(2k+1)!}(x^2)^{2k}=\sum^\infty_{k=0}\frac{(-1)^k}{(2k+1)!}x^{4k}$ Integrating, we have $F'(x)...
2
votes
3answers
81 views

Surprising Summation (4): $\frac 12 \sum_{i=1}^n (n+1-i)(n+i)=\sum_{i=1}^n i^2$

Show that $$\frac 12 \sum_{i=1}^n (n+1-i)(n+i)=\sum_{i=1}^n i^2$$ without expanding the summation to its closed form, i.e. $\dfrac 16n(n+1)(2n+1)$ or equivalent. e.g.for $n=5$, $$\frac12\bigg[5(...
0
votes
0answers
105 views

How to find power series with the following interval of convergence?

$a) [-1,1] $ (conditionally convergent both at $-1$ and $1$) $b) [e,\ \pi) $ $c)$ center at $x=-\sqrt{2}$ and interval of convergence $(-\infty, \infty)$ I think to solve the question, we basically ...
2
votes
2answers
29 views

Sum of reciprocals of squares - bounding

Recently in class our teacher told us about the evaluating of the sum of reciprocals of squares, that is $\sum_{n=1}^{\infty}\frac{1}{n^2}$. We began with proving that $\sum_{n=1}^{\infty}\frac{1}{n^2}...
6
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5answers
567 views

Can someone explain me this summation?

So I need to solve this summation $$\sum _{i=0}^{n-1}\left(\sum _{j=i+1}^{n-1}\left(\sum _{k=j+1}^{n-1}\:\left(1\right)\right)\:\right)$$ and this I know that the answer is $$\frac{n^3}{6}-\frac{n^2}{...
2
votes
2answers
49 views

Marsden's definition of Taylor Series

How does the following definition of Taylor polynomials: $f(x_0 + h)= f(x_0) + f'(x_0)\cdot h + \frac{f''(x)}{2!}h^2+ ... +\frac{f^(k)(x_0)}{k!}\cdot h^k+R_k(x_0,h),$ where $R_k(x_0,h)=\int^{x_0+h}...
1
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1answer
44 views

How to solve system of equations containing summation over variable to solve for?

How do we solve for $\pi_i$ in the following? $$\pi_i=\frac{\sum\limits_j N_{i,j}}{\sum\limits_j\left( \ell_{i,j} \frac{\sum\limits_k N_{k,j}}{\sum\limits_k \ell_{k,j} \pi_k}\right)}\qquad\forall i,\...
14
votes
2answers
166 views

Prove that $\sum_{n=1}^\infty \left(\phi-\frac{F_{n+1}}{F_{n}}\right)=\frac{1}{\pi}$

So, I know that $$\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\phi$$ where $F_n$ stands for the n'th Fibonacci number I was interested in measuring the error of the convergence of the above limit and was ...
0
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2answers
22 views

Writing Single summation with two variables

I have the following summation, What its the mathematically correct way to write this summation using the summation symbol? Note that I will need 2 variables.
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2answers
113 views

How do you use the Riemann Zeta Function?

I know that the Riemann Zeta Function is defined as: $$\zeta (s)=\sum_{n=1}^\infty \frac {1}{n^s}=\frac {1}{\Gamma (s)} \int _0^{\infty}\frac { x^{s-1}}{e^x-1} dx$$ Which I think would prove useful ...
0
votes
1answer
31 views

Solve for $x$ in a defined summation

I'm looking for the steps in solving for x in this summation: $$\sum_{i=1}^{12} 83.3(1+x)^i=1,100$$ I know that the solution is $x=0.0145764$, but I need to know how to arrive at the solution. My ...
0
votes
4answers
97 views

Show that $\sum_{i=n+1}^\infty \frac{1}{i^2-a^2} \approx \frac{1}{n}$

To show $\sum_{i=n+1}^\infty \frac{1}{i^2-a^2} \approx \frac{1}{n}$ where n is positive integer So far I have worked to: $=\frac{1}{2a}\cdot(-\sum_{i=n+1}^\infty \frac{1}{k+a}+\sum_{i=n+1}^\infty \...
2
votes
2answers
146 views

Simplifying this (perhaps) real expression containing roots of unity

Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ although I don't think that is relevant. Let $\zeta:=\exp(2\pi i/k)$ and $\alpha_v:=\zeta^v+\zeta^{-v}+\zeta^{-1}$. ...
0
votes
2answers
63 views

Some diffuculties trying to compute double sums

I have the following sum $$\sum_{i = 0}^{n-2}\sum_{j=i}^{n}(i + j) + \sum_{i = 0}^{n-2}\sum_{j=i}^{n}1$$? and i have no idea how to continue from here?
0
votes
1answer
17 views

Finding $n$ from the cumulative sum of the serie where $SUM(n) < \Pi < SUM(n+1)$

I have a serie of numbers: $$S = {1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/20, 1/30, 1/40, 1/50, 1/60, 1/70, 1/80, 1/90, 1/100, 1/200, 1/300, 1/400, 1/500, 1/600, 1/700, 1/800, 1/900,...}...
11
votes
3answers
142 views

Doubt regarding divisibility of the expression: $1^{101}+2^{101} \cdot \cdot \cdot +2016^{101}$

In an interesting contest question I recently encountered, I chanced upon a question I couldn't solve. $$\sum^{2016}_{i=1}i^{101}$$ is divisible by: (a)2014 (b)2015 (c)2016 (d)2017 How would I ...
1
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0answers
15 views

Sum of Gamma functions (product pairs)

This is my first time asking a question on stackexchange. Is there an analytical expression for the following summation of Gamma functions? $\sum_{t=0}^m \Gamma (A + t) \Gamma (B + m -t) = ?$ for ...
1
vote
2answers
77 views

Find the sum of the series: $\frac{1}{1*2} - \frac{1}{3*2^3} + \frac{1}{5*2^5} - \frac{1}{7*2^7}+\dots$?

$$\frac{1}{1*2} - \frac{1}{3*2^3} + \frac{1}{5*2^5} - \frac{1}{7*2^7}+\dots$$ I made a series to get $$\sum_{n=0}^\inf \frac{(-1)^n}{(1+2n)*2^{1+2n}}$$ but what series can it manipulate and simplify ...
2
votes
6answers
75 views

Is $\sum_{n=1}^{\infty} \frac{n-1}{n^2}$ convergent or divergent?

Is $\sum_{n=1}^{\infty} \frac{n-1}{n^2}$ convergent or divergent? I tried ratio test but didn't seem to work, and I also know that the limit goes to zero, but I can't say its convergence because then....