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

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0answers
32 views

Minimizing sum of weighted product

Consider a total of $d$ items, $\{I_1,I_2, \cdots,I_d\}$, each having a weight $w_i$ (a positive integer), and a total of $m$ bins, $\{B_1,B_2,⋯,B_m\}$. We would like to distribute the items into the ...
2
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5answers
84 views

Prove that $\left(\sum_{k=1}^{n}k\right)^2=\sum_{k=1}^{n}k^3$ holds true for $n ≥ 1$

I've been trying to figure out this proof for way too long now, I'm just not sure where to begin for the inductive step. Any guidance would be greatly appreciated.
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0answers
25 views

writing sum as a product and vice versa.

$\Pi = k$ from k = 1 to n Can you write this in form of sigma? So that you can evaluate it as a sum? Also, are there any shorthand formula to evaluate a product like there are for summations? ...
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1answer
40 views

Prove by Induction : $\sum n^3=(\sum n)^2$ [duplicate]

I am trying to prove that for any integer where $n \ge 1$, this is true: $$ (1 + 2 + 3 + \cdots + (n-1) + n)^2 = 1^3 + 2^3 + 3^3 + \cdots + (n-1)^3 + n^3$$ I've done the base case and I am having ...
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1answer
24 views

MacLaurin of the Third-degree in sin(a*x)*cos(b*x) at given values

Alright so from my understanding MacLaurin is a special case of Taylor Series but at f(0). However my question involves solving the third degree of MacLaurin for $$f(x) = sin(a \times x)\times ...
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1answer
72 views

Evaluating Telescopic Sum $ \sum\frac{n}{1+n^2+n^4} $

How to evaluate following $$ \sum_{n=1}^{\infty}\frac{n}{1+n^2+n^4}$$ I posted my way as an answer, Is there another Interesting approach to evaluate this sum of series?
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1answer
22 views

differential equation using series expansion

Trying to solve xy'= xy + y using the series y(x) = $\sum\limits_{i=0}^\infty a_nx^n$ This is what i have so far. y'(x)= $\sum\limits_{i=0}^\infty na_nx^{n-1}$ xy' - xy - y = 0 x ...
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2answers
43 views

What is the general formula for power series summation? [duplicate]

While reviewing definite integrals, $\int_a^bf(x)dx$; I recalled that a definite integral could not only be solved by the difference of the anti-derivatives of intervals b and a, $F(b)-F(a)$, via the ...
0
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1answer
27 views

Derivatives and Integrals of Summations

Im unsure if this is just a stupid question because i have been independently studying this kind of math for about a week, but this has been bothering me lately as i have been exploring some definite ...
7
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1answer
111 views

The meaning of a definition involving multiple sums with Bernoulli numbers

Reading a paper regarding Bernoulli numbers, and I stumbled onto a definition. First let $$\frac{x}{e^x-1}=\sum_{k=0}^{\infty}B_k\frac{x^k}{k!}$$ The author then goes on to define new terms. Let ...
2
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1answer
30 views

$\sum_{i=0}^n (-1)^i \delta_i(\sum_{j=0}^{n+1} (-1)^j \delta_j) = 0$, given that $\delta_i \delta_j = \delta_{j-1}\delta_i$ whenever $i < j$

$\sum_{i=0}^n (-1)^i \delta_i(\sum_{j=0}^{n+1} (-1)^j \delta_j) = 0$, given that $\delta_i \delta_j = \delta_{j-1}\delta_i$ whenever $i < j$ This problem shows up in the middle of dealing with ...
4
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1answer
27 views

How to tell which function's Fourier series to use in order to calculate the value of series.

I got this question when I was doing some exercises. I was ask to establish $$ \sum_{n=0}^{\infty}\frac {1}{(2n+1)^2}=\frac{\pi^4}{96},\quad \sum_{n=0}^{\infty}\frac ...
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3answers
22 views

Calculating sum of consecutive powers of a number

Here is my problem, I want to compute the $$\sum_{i=0}^n P^i : P\in ℤ_{>1}$$ I know I can implement it using an easy recursive function, but since I want to use the formula in a spreadsheet, is ...
6
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2answers
112 views

What is the value of $\sum_{m=1}^{19} \frac{1}{\zeta^{3m}+\zeta^{2m}+\zeta^{m}+1}$ with $\zeta=e^{2\pi i/19}$?

Given that $\zeta=e^{2\pi i/19}$, how to find the value of $$S=\sum_{m=1}^{19} \dfrac{1}{\zeta^{3m}+\zeta^{2m}+\zeta^{m}+1}$$? All I could think of was to somehow factorize the denominator and apply ...
8
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2answers
84 views

Exactly expressing integral as a sum

Apparently (i.e. according to my professor), the following holds:$$\int_a^b f(x) dx = (b-a)\sum_{n=1}^\infty \sum_{m=1}^{2^n-1} (-1)^{m+1}2^{-n}f(a+m(b-a)2^{-n}).$$How would one go about proving such ...
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2answers
38 views

Is this sum equivalent?

Does $\sum_{n=1}^\infty -\frac{4\cos(nx)}{10n^3 \pi}\cos(nt)\sin(nx)=\frac{2}{n^3}\cos(nt)\sin(nx)(1-\cos(nx))$? I don't think it does. How can I check?
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0answers
28 views

Uniform convergence of function

Consider $$f_C(x)=\sum_{n=2}^\infty \frac{\cos(2\pi nx)}{n\log n}$$ Show that it does not converges uniformly and $f_C(x)\geq c\log \log \frac{1}{|x|}$ as $x\rightarrow 0$. I used summation parts ...
4
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1answer
37 views

How can I find a closed form for the summation (i^2)(-1^i+1) systematically?

In one of my homeworks I was given the following sequence $1^2-2^2+3^2-4^2+\dots (-1)^{n+1}n^2$, and I'm supposed to find a closed form formula and prove that it works. Rewriting this as a sum gives ...
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2answers
82 views

Growth of $\sum_{x=1}^{n-1} \left\lceil n-\sqrt{n^{2}-x^{2} } \right\rceil$

I'm interested in the growth of $$f(n):=\sum_{x=1}^{n-1} \left\lceil n-\sqrt{n^{2}-x^{2} } \right\rceil \quad \text{for}\quad n\rightarrow\infty $$ Progress (From comments) I've got ...
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0answers
22 views

How to factor $(1-(i^2/n)(1/n)$ to isolate $i^2$ and form a sigma identity?

given sigma from $i=1$ to $n$ of $(1-(i^2/n)^2)(1/n))$ how would you factor this function to isolate $i^2$ and get $[n(n+1)(2n+1)]/6$ ? update... I got until the limit as n approaches infinity (1/n) ...
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1answer
59 views

Mathematical induction base case is not initial

Prove by induction that $$1+2+3+\cdots+n= \frac{n(n+1)}{2}$$ for all integers greater than or equal to $2$ How can you solve this if the base case is not $1$? I thought it might be a strong ...
1
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1answer
71 views

Find summation of following series.

What will be the formula for following infinite series? $$1 + \frac{1!}{x+1} + \frac{2!}{(x+1)(x+2)}+ \cdots$$ $$ x\ge2 $$ up to infinite What pattern i got : coefficient of $ \frac{1!}{x+1}$ ...
5
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3answers
136 views

How to calculate convergence of this function?

Given $n$, is there any easy way to calculate convergence of this summation. $$\sum_{k=0}^\infty\dfrac{1}{^{n+k}C_n}$$ EDIT: Also I need to find at which value this series converges.
5
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1answer
90 views

How to prove $\sum_{n=0}^{\infty} \frac {(2n+1)!} {2^{3n} \; (n!)^2} = 2\sqrt{2} \;$?

I found out that the sum $$\sum_{n=0}^{\infty} \frac {(2n+1)!} {2^{3n} \; (n!)^2}$$ converges to $2\sqrt{2}$. But right now I don't have enough time to figure out how to solve this. I would ...
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6answers
224 views

Prove that $\sum_{n=1}^\infty \frac{n^2(n-1)}{2^n} = 20$

This sum $\displaystyle \sum_{n=1}^\infty \frac{n^2(n-1)}{2^n} $showed up as I was computing the expected value of a random variable. My calculator tells me that $\,\,\displaystyle ...
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4answers
38 views

Understanding Summation sequence

I'm trying to wrap my head around this summation. I understand basic ones for the most part, stop, start, etc, but I don't understand this one in particular.
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2answers
36 views

To prove $\sum\limits_{k=1}^{n}k \cos{\frac{2k\pi}{n}} =-\frac {n}2$

Prove that $\sum\limits_{k=1}^{n}k \cos{\frac{2k\pi}{n}} =-\frac {n}2$, for $n\in\mathbb{Z}, n\ge3$
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1answer
56 views

Summing an infinite series

I have been struggling with a problem involving a Markov Chain. To solve it I need to figure out the following ...
2
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0answers
54 views

Evaluating a limit…

I was solving a physics problem and this expression came about: $E =\lim_{N \to \infty} \left( \dfrac{k_0Q}{NR²}\displaystyle\sum_{i=0}^{(N/2-1)}\left[ \left( ...
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1answer
28 views

Question about sums with a negative limit for the index

To me, it looks like we have $\;\sum_{i = 1}^{0} x_i = 0\;$ and $\;\sum_{i = 1}^{1} x_i = x_1\;$. What happens if I write the following? $$\;\sum_{i = 1}^{-123} x_i\;$$ Would this be defined?
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2answers
85 views

Is $\sum i^{1/i}$ bounded?

I'm trying to find the limit $$ \lim_{n\to\infty}\sum_{i=1}^n \frac{i^{1/i}}n\,. $$ I was going to say that $\lim_{n\to\infty} \frac1n=0$ and $\sum i^{1/i}$ is bounded but I can't prove it.
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0answers
29 views

n integrals in summation

I've seen it might be possible to write a summation that looks like $$ \sum\limits_{i=1}^{\infty}\left\{\frac{\partial}{\partial x_i}\left(\frac{xy}{\sqrt{x^2+y^2}}\right)\right\} $$ But what about ...
0
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1answer
36 views

Sum of numbers in arithmetic series [closed]

I need some help to find the summation of the series of following from $\frac{1}{(n+10)^2}$. I need to get the some from $n=0$ to $n=2280$. can anybody help me find the answer for this. Thanks in ...
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1answer
28 views

How to prove that Grandi's series $= \frac{1}{2}$ using Euler transform [closed]

Let $x$ denote Grandi's series $1-1+1-1+1-1+1-...$ This implies that $$ x = 1\text{ or}\\ x = 0\text{ or}\\ 1-x = 1 - (1-1+1-1+1-...) = x \implies 2x = 1 \implies x = \frac{1}{2}$$ Where the last ...
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2answers
40 views

Show that $\sum_{k=1}^{p-1}f(k)=\sum_{k=1}^{p-1}f(qk)$

I would appreciate if somebody could help me with the following problem Q: Show taht $$\sum_{k=1}^{p-1}f(k)=\sum_{k=1}^{p-1}f(qk)$$ where $\gcd(p,q)=1, f(p+x)=f(x)$
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1answer
40 views

Help with derivative inside a summation

I have $\sum_{k=0}^{\infty}k^2q^kp=\sum_{k=0}^{\infty}k[kq^{k-1}]qp=\sum_{k=0}^{\infty}k[\frac{d}{dq}(q^k)]qp$. How can I go about pulling this $\frac{d}{dq}$ outside of the sum?
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2answers
53 views

Rewriting an infinite sum

Rewrite the given expression as a sum whose generic term involves x^n: $$ x\cdot\sum_{n=1}^{\infty}(n a_n x^{n-1}) + \sum_{k=0}^{\infty}(a_k x^{k} ) $$ I get a sum starting at one: $$ ...
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2answers
70 views

Help finding value of N that minimizes a sum

Suppose we have the following inequality: $\sum\limits_{k=N+1}^{1000}\binom{1000}{k}(\frac{1}{2})^{k}(\frac{1}{2})^{1000-k} = \frac{1}{2^{1000}}\sum\limits_{k=N+1}^{1000}\binom{1000}{k} < ...
9
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6answers
282 views

Evaluating $ \sum\frac{1}{1+n^2+n^4} $

How to evaluate following expression? $$ \sum_{n=1}^{\infty}\frac{1}{1+n^2+n^4}$$ I doubt it is a telescopic Sum.
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2answers
50 views

Compute $\sum _{i=0}^n \left(\frac{i}{n}\right)^3=\frac{(n+1)^2}{4 n}$

Can someone show me how one can deal with this get the answer provided? $$\sum _{i=0}^n \left(\frac{i}{n}\right)^3=\frac{(n+1)^2}{4 n}$$ Thanks
1
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1answer
38 views

$\sum _1 ^n |z_j| \ge 1 \Rightarrow | \sum _1 ^k z_{j_m}| \ge C$

Prove that there exists $C > 0$ such that the following implication holds: If $\{z_1, ..., z_n \} \subset \mathbb{C}$ are such that $\sum _{j=1} ^n |z_j| \ge 1$, then there exists $ \{z_{j_1}, ...
2
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1answer
34 views

Evaluation of a limit of ratio of sums [closed]

How do I calculate the value of $$ \lim_{n\to \infty} \left(\frac{\sum_{r=0}^{n} \binom{2n}{2r}3^r}{\sum_{r=0}^{n-1} \binom{2n}{2r+1}3^r}\right)$$
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0answers
18 views

help in simplifying an easy but nasty expression

I would like double check my work, I am trying to simplify the following summation, \begin{align} \sum_{\substack{(i,j) \in \mathcal{S}}} A_i v_i v_j \end{align} with the assumption that $$v_iv_j= c ...
2
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4answers
280 views

Sum notation confusion

Consider the following sum: $$\sum_{i=0}^n e^{i/n}$$ I don't understand this notation. Apparently the closed form is $$\dfrac{e^{(n+1)/n }-1}{e^{1/n} -1}$$ But it says $i=0$. I really don't ...
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0answers
42 views

Find $ \sum\limits_{n=0}^\infty \frac{2^n}{3^nn!}$

Find $\displaystyle \sum_{n=0}^\infty \frac{2^n}{3^nn!}$
3
votes
1answer
70 views

How to find asymptotics of this sum

Is there any way to find $f(n)$ in this term: $$\sum_{k=2}^n \frac1{\ln \ln(k!^{k!})} \sim f(n)?$$ The tilde symbol means that $$\lim_{n\to∞} \frac{f(n)}{\sum_{k=2}^n \frac1{\ln \ln(k!^{k!})}} = 1$$ ...
6
votes
5answers
129 views

Verify the following combinatorial identity: $\sum_{k=0}^{r} \binom{m}{k}\binom{n}{r-k} = \binom{m+n}{r}$

$$\sum_{k=0}^{r} \binom{m}{k}\binom{n}{r-k} = \binom{m+n}{r}$$ Nice, so I've proven some combinatorial identities before via induction, other more simple ones by committee selection models.... But ...
0
votes
1answer
41 views

How to write in closed form this nasty expression?

I have something like $$ v_1 l_1+v_1 l_2+ v_2l_1+v_2l_2$$ and I am trying to write it in closed form as such, $$\sum_{j=1}\sum_{i=1}v_il_j$$ I know this is not right but I want something like that. ...
4
votes
2answers
70 views

Show that $k^a=\sum_{m=1}^b\left ( c_m^a\prod_{n\neq m} \frac{k-c_n}{c_m-c_n} \right ).$

I used the following result in another post without providing proof (because I couldn't prove it): $$k^a=\sum_{m=1}^b\left ( c_m^a\prod_{n\neq m} \frac{k-c_n}{c_m-c_n} \right ),$$ where $a$ and $b$ ...
1
vote
0answers
47 views

“Balancing” Sums

Given are $x_1,\ldots, x_n\in \{0,1,\ldots,n\}$, $y_1,\ldots, y_n\in \{0,1,\ldots,n\}$ with the property that $$\sum_{i=1}^{n}{x_i}\leq B,$$ $$\sum_{i=1}^{n}{y_i}\leq B$$ Let's assume that $B$ is ...