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

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5
votes
2answers
127 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 ...
7
votes
6answers
247 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 ...
0
votes
4answers
41 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.
0
votes
2answers
38 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 $n\in\mathbb{Z}, n\ge3$
1
vote
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
votes
0answers
56 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( ...
0
votes
1answer
29 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?
0
votes
2answers
87 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.
0
votes
0answers
31 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
votes
1answer
37 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 ...
1
vote
1answer
31 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 ...
2
votes
2answers
44 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 $\gcd(p,q)=1, f(p+x)=f(x)$
0
votes
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?
0
votes
2answers
79 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: $$ ...
1
vote
2answers
72 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} < ...
10
votes
6answers
319 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.
2
votes
2answers
54 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
vote
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
votes
1answer
41 views

Evaluation of a limit of ratio of sums [closed]

How do I calculate the value of $$ \lim_{n\to \infty} \left(\frac{\sum_{r=0}^{n} \binom{2n}{2r}3^r}{\sum_{r=0}^{n-1} \binom{2n}{2r+1}3^r}\right)$$
0
votes
0answers
20 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
votes
4answers
289 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 ...
3
votes
1answer
87 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
157 views

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

$$\sum_{k=0}^{r} \binom{m}{k}\binom{n}{r-k} = \binom{m+n}{r}$$ Nice, so I've proven some combinatorial identities before via induction, other more simple ones by committee selection models.... But ...
0
votes
1answer
45 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
73 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
51 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 ...
0
votes
1answer
72 views

Simplify sum notation with factorial and exponent [closed]

How would I simplify this to an equation: $$\lim_{m \to \infty}\sum_{k=0}^{m-1}\frac{x^k}{k!}$$
0
votes
1answer
21 views

How to eliminate coefficients from a sum

For given random values $$X_i \sim\mathcal{N}(0,1)$$ and $$\frac{X_i-\mu}{\sigma}=\tilde{X_i}\sim\mathcal{N}(\mu,\sigma),\,\mu\in\mathbb{R},\,\sigma>0$$ prove ...
0
votes
1answer
41 views

Sum of positive integers estimating sum of fractions

Given $m$ fractions adding up to an positive integer $n$ For example: $m=3\\n=10=\frac{30}{6}+\frac{20}{6}+\frac{10}{6}$ How can we find $m$ positive integers that sum to $n$ (a partition of $n$), ...
1
vote
1answer
73 views

Finding the asymptotics of $\sum_{k=1}^n a^k k!$? Note that $a > 0$.

There's no way to use integration method in this case. I also tried to use Stolz–Cesàro theorem, but couldn't find right $y_n$. What method should I use?
0
votes
3answers
35 views

How does an index greater than 1 affect this arithmatic series?

I was wondering how the starting index affects this arithmatic series and series in general when using the formulas: $\displaystyle=2\times\left(\frac{41\times40}2\right)-1$ $\displaystyle=1640-1$ ...
1
vote
1answer
57 views

prove that $ \binom n 2 + \binom {n-2} 2 + \binom {n-4} 2 + \dots + \binom 3 2 = \frac 1 {24} (n-1)(n+1) (2n+3) $

$$ \binom n 2 + \binom {n-2} 2 + \binom {n-4} 2 + \dots + \binom 3 2 = \frac 1 {24} (n-1)(n+1) (2n+3) $$ where n is odd. Plesase help mi with that equation.
5
votes
2answers
109 views

How find this sum $\frac{1}{1^2}+\frac{2}{2^2}+\frac{2}{3^2}+\frac{3}{4^2}+\frac{2}{5^2}+\frac{4}{6^2}+\cdots+\frac{d(n)}{n^2}+\cdots$

Question: Find the value $$\dfrac{1}{1^2}+\dfrac{2}{2^2}+\dfrac{2}{3^2}+\dfrac{3}{4^2}+\dfrac{2}{5^2}+\dfrac{4}{6^2}+\cdots+\dfrac{d(n)}{n^2}+\cdots$$ where $d(n)$ is The total number of ...
3
votes
0answers
52 views

Random Wolfram|Alpha identity $\sum_{k = 1}^{\infty}{\tan^{-1}}{\frac{1}{k^2}}$

I was watching a Numberphile video (on how $\tan^{-1}{1} + \tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{3} = \frac{\pi}{2}$) and I thought about whether the series $$\sum_{k = ...
2
votes
2answers
116 views

general way to get formula as multiplication.

We assume: $$ 1^n + 2^n + 3^n + .. + k^n$$ where k and n are natural numbers. Are there a general way to get it as multiplication? For example: $$ 1^3 + 2^3 + 3^3 + .. + k^3 = \binom {k+1} 2 ^2 $$
1
vote
1answer
56 views

Can the absolute value of a sum be expressed as the sum of absolute values? [closed]

If I have $y_i=\sum_{j=1}^i P_{ij}x_j$ can I say that's equivalent to $|y_i-x|\le\sum_{j=1}^i P_{ij}|x_j-x|$
0
votes
3answers
41 views

Help explaining sum notation

So, I have the question and I also have the answer. Need to prove: And here is the answer Can you please explain the steps. They are in the second picture but I do not understand where they are ...
1
vote
0answers
25 views

How do I convert this equation from Iverson brackets to make use of the Heaviside function?

I have the equation $\sum_{i=0}^{\infty} 2^{i}[0 \leq x - 2^{i}][x - 2^{i + 1} < 0]$ and I would like to convert the Iverson brackets to the Heaviside function. I've read this post but I'm ...
2
votes
1answer
60 views

Calculate $\sum_{k=51}^{\infty} \frac{(-1)^{k-1}}{k} $

I have a sum to calculate: $$\sum_{k=51}^{\infty} \frac{(-1)^{k-1}}{k} $$ And I have no idea about how to proceed. What kind of techniques are available to calculate this?
1
vote
4answers
39 views

Summation of polynomial expression

I'm looking for some assistance to the following problem: $$\sum_{k=3}^{n}(k^2 - 3) = \sum_{k=3}^{n}{k^2} - \sum_{k=3}^{n}{3}$$ $$= \frac{n(n+1)(2n+1)}{6} - 3n$$ However, I know the last term is ...
0
votes
2answers
20 views

Arithmetic sequence problem

I'm having some trouble finding the best approach to the following arithmetic sequence problem: $$\sum_{i=3}^{30}{[(i-3)^2+i-3]}$$ I'm aware that I can break up the sequence: ...
6
votes
6answers
237 views

Finding $\sum_{k=1}^{\infty} \left[\frac{1}{2k}-\log \left(1+\frac{1}{2k}\right)\right]$

How do we find $$S=\sum_{k=1}^{\infty} \left[\frac{1}{2k} -\log\left(1+\dfrac{1}{2k}\right)\right]$$ I know that $\displaystyle\sum_{k=1}^{\infty} \left[\frac{1}{k} ...
0
votes
2answers
80 views

Solve $ \sum_{k = 1}^{ \infty} \frac{\sin 2k}{k}$

Solve $$ \sum_{k = 1}^{ \infty} \frac{\sin 2k}{k}$$ I first tried to use Eulers formula $$ \frac{1}{2i} \sum_{k = 1}^{ \infty} \frac{1}{k} \left( e^{2ik} - e^{-2ik} \right)$$ However to use the ...
2
votes
0answers
141 views

The history of summations

How did summations evolve? For instance, is there an article, book, webpage, etc. that talks about how mathematicians came up with using $\sum_x{ f(x) }$? I'm very interested on how summations came ...
0
votes
2answers
21 views

Summing powers of complex root of unities gives 0

I have a question regarding a proof. Let $z_N$ denote the complex N'th root of unity, from which we have the identities $(z_N)^n=1$ $\sum_{i=0}^{N-1}{(z_N)^i}=0$ Now let $N=r\cdot t$ and let ...
0
votes
2answers
26 views

Probing a particular function

I've been playing with a particular function $$Q(n) = \sum_{i=1}^n i\cdot i!$$ in C++, and I'm trying to see if it is possible to find the following in an elegant way: 1) Is it possible to rewrite ...
2
votes
2answers
102 views

Closed form for this summation

How do we prove that $$S=\displaystyle\sum_{n=1}^{\infty} e^{-n} \sin n=\dfrac{e\sin 1}{1+e^2-2e\cos 1}\approx0.419$$ We can write the sum as $$S=\Im \sum_{n=1}^{\infty} e^{(i-1)n}$$ I do not know ...
0
votes
0answers
39 views

problems with applying a $f(x)=x^2$ curve

I am applying a curve over time (two seconds), to transition from one value to another. The formula I am using is: $$x = \left(\frac {\text{time}}{2.0}\right)^2$$ ...
0
votes
3answers
58 views

Can a sum of products be split as a product of two sums?

I have $$\sum_k^n P_k x_k$$ Am I allowed to split it up into two sums so I have it like $$\sum_k^n P_k \sum_k^nx_k$$
0
votes
1answer
44 views

Confusion when applying Tensor transformation law to $\partial_{[a,v_b]}$

What I'm trying to show is that, if $v_a$ is a covector field, $\partial_{[a, v_b]} = \frac{1}{2}(\partial_a v_b - \partial_b v_a)$ transforms like a type $(0,2)$ tensor. First of all, a type ...