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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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
36 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
67 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
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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: $\displaystyle=2\times\left(\frac{41\times40}2\right)-1$ $\displaystyle=1640-1$ ...
-2
votes
1answer
60 views

What does $\sum_{i = 2}^{\infty} \frac{1}{i(i-1)}$ converge to? [closed]

What does $\sum_{i = 2}^{\infty} \frac{1}{i(i-1)}$ converge to? That is, $\frac{1}{2(1)}+\frac{1}{3(2)}+ \frac{1}{4(3)} +... $
1
vote
1answer
56 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
97 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
51 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 = ...
0
votes
1answer
52 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 $|y_i-x|\le\sum_{j=1}^i P_{ij}|x_j-x|$
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3answers
38 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
10 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
59 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
36 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 ...
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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: ...
5
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6answers
202 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
79 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
89 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 ...
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votes
2answers
20 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
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2answers
25 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
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2answers
99 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 ...
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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
46 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
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1answer
32 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 ...
1
vote
1answer
47 views

Proof that sum of power series equals exponential function?

I have found that the Sum series equal an exponential function as below, however I have not found a proof for it: $$ ze^z = \sum_{k=0}^{\infty} k \frac{z^k}{k!} $$ I have though managed to prove ...
0
votes
1answer
28 views

Multivariable calculus, inner products

I am trying to solve this question. I have considered ith component and replaced it with $v_i/(v_i^2)^{1/2}$ and the summation form of the dot product, but cannot see how the RHS falls out, can ...
0
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1answer
33 views

Replace $\sum$ with $\ln$

Reading an solution I get stuck at this step. $$ \frac{1}{2i} \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n} \left( e^{2in} - e^{-2in} \right) \\ = - \frac{1}{2i} \left( \ln(1 + e^{2i}) - \ln(1- e^{-2i}) ...
2
votes
2answers
40 views

Sums with squares of binomial coefficients multiplied by a polynomial

It has long been known that \begin{align} \sum_{n=0}^{m} \binom{m}{n}^{2} = \binom{2m}{m}. \end{align} What is being asked here are the closed forms for the binomial series \begin{align} S_{1} &= ...
1
vote
1answer
34 views

Finding a Derivation for a particular Series

This series came up when working out a physics problem, and I have been unable to derive the sum with any rigor. Here is the series and it's probable evaluation... $$\sum_{odd m\neq ...
11
votes
5answers
141 views

Evaluate $\frac{ 1 }{ 1010 \times 2016} + \frac{ 1 }{ 1012 \times 2014} + \frac{ 1 }{ 1014 \times 2012} + \cdots + \frac{ 1 }{ 2016 \times 1010} = ?$

$$\dfrac{ 1 }{ 1010 \times 2016} + \dfrac{ 1 }{ 1012 \times 2014} + \dfrac{ 1 }{ 1014 \times 2012} + \cdots + \dfrac{ 1 }{ 2016 \times 1010} = ? $$ My attempt so far : ...
0
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0answers
53 views

What is the sum of 1^2 + 2^2 + 3^2… + T^2? [duplicate]

I need to find the sum of this sequence. Thank you in advance
1
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0answers
52 views

A challenge question in elementary number theory!

Find an expression for the following sum: $$\sum_{i:(i,n)=1}(i-1,n)$$ I guess that this sum equals to $\phi(n)d(n).$
1
vote
1answer
43 views

Is there a name and/or notation for arbitrary sum of sums?

I am looking for information about sums of the form: $$\sum_{i=1}^n \sum_{j=1}^i f(j)$$ But not just that form, but arbitrarily many stacked sums. Even just a name would help. To be specific about ...
4
votes
4answers
125 views

Calc the sum of $\sum_{k = 0}^{\infty} \frac{(-1)^k}{k} \sin(2k)$

Solving a bigger problem about Fourier series I'm faced with this sum: $$\sum_{k = 0}^{\infty} \frac{(-1)^k}{k} \sin(2k)$$ and I've no idea of how to approach this. I've used Leibniz convergence ...
0
votes
3answers
50 views

Alternating sum with binomial coefficients

$\sum_{k=0}^{49}(-1)^k\binom{99}{2k}$ = ? I've tried expanding the binomial coefficient in its factorial form and can't seem to get to manipulate it in a way that solves the expression. ...
2
votes
1answer
48 views

Strong induction on a summation of recursive functions (Catalan numbers)

I've been stuck on how to proceed with this problem. All that's left is to prove this with strong induction: $$\forall n \in \mathbb{N}, S(n) = \sum_{i=0}^{n-1} S(i)*S(n - 1 - i)$$ Some cases: S(0) ...
0
votes
2answers
41 views

How to take this derivative

My question is straightforward: I need to evaluate an expression of the form $$ \frac{\partial}{\partial t}\sum_{k=0}^{t}\varphi(k,t) $$ How is this done, usually?
0
votes
1answer
29 views

closed form of summation of geometric theories

I am working on a discrete convolution problem. I am comparing the solution to what the solution manual has and it just doesn't make sense to me. what solution manual has: http://imgur.com/lllGPmG ...
0
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1answer
49 views

Is it true that $\Big(\sum_{n=1}^N a_n\Big)^p\leq\sum(a_n)^p$?

More precisely, is the inequality $\Big(\displaystyle\sum_{n=1}^N a_n\Big)^p\leq\sum(a_n)^p$ true for $a_n\geq0$ for all $n\in\{1,\ldots,N\}$ and $p\in(0,\infty)$? EDIT: And if so, will it also hold ...
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2answers
46 views

series of $\arctan \frac{2x^2}{n^2}$ .

Find : $$\sum_{n=1}^{\infty}\arctan \frac{2x^2}{n^2}.$$ Where $x\in \mathbb{R}$.
1
vote
1answer
24 views

Proving that the multiplication on a formal derivation ring is associative

Let $R$ be a ring , $\delta$ a derivation of $R$, and $x$ an indeterminate. Let $R[x,\delta]$ be for the formal differential operator ring over $R$ (so for $a \in R$, $xa = ax + \delta(a)$). ...
3
votes
2answers
62 views

Calculate the sum of infinite series with general term $\dfrac{n^2}{2^n}$. [duplicate]

Please explain different methods to calculate the sum of infinite series with $\dfrac{n^2}{2^n}$ as it's general term i.e. Calculate $$\sum_{n=0}^\infty \dfrac {n^2}{2^n}$$ Please avoid the method ...
0
votes
0answers
20 views

sum polylogarith convergence

I am working in the following series i get a result could you help to see the convergence any ideas $$\sum _{k=1}^{\infty } \frac{(-1)^{k+1} \text{Li}_k(-a)}{a}=\frac{a-\log (a+1)}{a^2}$$ where Lk is ...
3
votes
3answers
56 views

Proof of Equation by Well Ordering Principle

I have an assignment question Prove by either the Well Ordering Principle or induction that for all nonnegative integers $n$: $$\sum_{k=0}^n k^3 = \left(\frac{n(n+1)}{2}\right)^2.$$ I am able to ...
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2answers
33 views

Rewrite $\sum_{n\ge0}{z^{kn}}$ to $\sum_{n\ge0}{f(n,k)z^n}$

In the simplest case: $\sum_{n\ge0}{z^{2n}}=\sum_{n\ge0}{\frac12((-1)^n+1)z^n}$. How to express $f(n,k)$ in closed form? If it's intractable, how to avoid piecewise expression?
0
votes
0answers
14 views

How can you find the integer part of y starting from this inequality?

How can you find the integer part of y starting from this equality? (I need a precise procedure, not only the number)
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2answers
35 views

Partial sum formula of a polynomial series?

I am trying to find the partial sum formula of the following series: $$ \sum_{y=1}^{\infty} \frac{4y^2-12y+9}{(y+3)(y+2)(y+1)y} $$ I have tried using Faulhaber's formula without success. I have also ...
0
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2answers
33 views

Finding a sum of alternating cubes

Find this sum, in terms of $n$: I have some hand-written hints from someone else, but can't read his writing:
2
votes
2answers
104 views

Estimating sum with binomial coefficients

Lately when I was estimating complexity of some algorithm I came across this sum: $$\sum_{k=0}^n \binom {n}{k} \binom {n-k}{k}$$ Is it possible to find a closed-form expression for this sum, or at ...
2
votes
1answer
64 views

Find the closed form of an n-sum

I'd like to find the closed form or a quickly converging rewriting of the following n-sum: ...
1
vote
1answer
34 views

Infinite sum question

How to solve the following infinite sum: $$\sum_{n=0}^\infty \frac{[(-25)^n+(-9)^n+(-1)^n]x^{(2n)}}{(2n)!}$$ I have no idea where to go from here, but I began with $\cos(5x) + \cos(3x) + \cos(x)$. I ...