Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.
0
votes
1answer
41 views
Finding expected value of variance estimator (sum expansion problem)
I am trying to show that variance estimator $\frac{1}{n}\sum_{i=1}^{n}(X_i-\bar{X})^2$ is biased. I have an example in the book, and there is one step of this derivation I cannot understand:
...
2
votes
2answers
91 views
Interval of convergence of $\sum_{n=4}^\infty x^n/n^5$
Find the interval of convergence $$\sum_{n=4}^\infty x^n/n^5$$
I'm lost here. My intuition was to use the ratio test.
$$\lim_{n \to \infty} \frac{x^{n+1}}{(n+1)^5} \times \frac{n^5}{x^n} $$
...
0
votes
1answer
49 views
Manipulating double summation
In a problem in my book, the following equality is there:
$$\sum_{n=0}^\infty\Big( \sum_{k_i\ge 0, \sum_{i=1}^\infty ik_i=n}\frac{x^n}{\prod_{i=1}^\infty k_i!(i!)^{k_i}}\Big)=\sum_{k_i\ge ...
1
vote
0answers
55 views
Double sum with binomial coefficients
Find a closed form formula for this sum:
$$\sum_{1\le i<j\le m} \sum_{\substack{1\le k,l\le n \\ k+l\le n}}{n\choose k} {n-k\choose l} (j-i-1)^{n-k-l}$$
It's quite likely that it can be ...
1
vote
4answers
88 views
What is the formula of: $a^{0} + a^{1} + a^{2} + … + a^{n-1} + a^{n}$? [duplicate]
What is the formula of:
$$a^{0} + a^{1} + a^{2} + ... + a^{n-1} + a^{n}$$
Any ideas?
4
votes
4answers
139 views
Proving that $\sum_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$
I have to prove that:
$$\sum_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$$
I tried to open up the right side with Pascal's definition that:
$$ { n \choose k} = {n-1 \choose {k}} + {n-1 ...
0
votes
1answer
24 views
Showing an indentity with a cyclic sum
let $z_1,z_2,..,z_n$ be non equal complex numbers
for any $n\geqslant2$, for any $k\in \mathbb{N}$
$$ \sum_{i=1}^{n}\frac {{z}_{i}^{n-1+k}} { \prod \limits_{\substack{j = 1\\j \ne i}}^{ n }{ ...
0
votes
2answers
43 views
Proof of functional equation
I have a function
$$ g_N(x) =\frac{1}{x}+ \sum_{n=1}^\infty \Big(\frac{1}{x+n}+\frac{1}{x-n}\Big) $$
How can I prove that $$g_N(\frac{x}{2})+g_N(\frac{x+1}{2}) = 2g_N(x)\;?$$
2
votes
4answers
41 views
Equality proof of function $g_N(x) = \sum_{n=-N}^N \frac{1}{x+n}$
I have a function
$$ g_N(x) = \sum_{n=-N}^N \frac{1}{x+n} $$
How can I prove that this function is odd, thus $ g_N(-x) = -g_N(x)$ ?
1
vote
2answers
59 views
Finding an infinite trigonometric sum
Find the following infinite sum : $$q\sin a+q^2\sin 2a+\ldots+q^n\sin na+\ldots$$ where $|q|<1$ .It would be good if you could find it without the help of any auxiliary sequences using only ...
1
vote
0answers
53 views
Azuma's inequality with high probabilistic bounds
Let $(X_n)_{n \geq 0}$ be a super-martingale, that is $\mathbb{E}[X_{n+1} \mid X_1, \dots, X_n] \leq X_n$. Let's further assume that $\Pr[|X_n - X_{n-1}| < c_n] \geq 1-\delta$. Does there exist any ...
-1
votes
1answer
65 views
Sum with Stirling numbers
Show that for each $n>1$
$$\sum\limits_{k=1}^{n-1} \frac{(n-1)!}{(k-1)!}S(n,n-k) = (n-1)^n $$
where $S(n,m)$ is the Stirling number of the second kind.
1
vote
1answer
44 views
A finite sum of trigonometric functions
By taking real and imaginary parts in a suitable exponential equation, prove that
$$\begin{align*}
\frac1n\sum_{j=0}^{n-1}\cos\left(\frac{2\pi jk}{n}\right)&=\begin{cases}
1&\text{if } k ...
20
votes
3answers
410 views
Finding the value of $\sum\limits_{k=0}^{\infty}\frac{2^{k}}{2^{2^{k}}+1}$
Does this weighted sum of reciprocals of Fermat numbers,
$$
F=\sum_{k=0}^{\infty}\dfrac{2^{k}}{2^{2^{k}}+1}
$$
have a nice closed form? Wolfram says it's $1$.
Thanks.
7
votes
1answer
169 views
Another Series $\sum\limits_{k=2}^\infty \frac{\log(k)}{k}\sin(2k \mu \pi)$
I ran across an interesting series in a paper written by J.W.L. Glaisher. Glaisher mentions that it is a known formula but does not indicate how it can be derived.
I think it is difficult.
...
0
votes
1answer
37 views
sum of series $\sum_{n=0}^\infty \left(\frac1{1+3\cdot(-1)^n}\right)^n$
Not sure how to evaluate this. I'm pretty sure I should examine the difference between even and odd indicies.
So it's jumping between:
$(1/4)^n$ when $n$ is even
and $(-1/2)^n$ when $n$ is odd
...
3
votes
0answers
32 views
Proving two summations equivalent [duplicate]
Let $h_n$ be an infinite sequence. I need to show that:
\begin{align}\dfrac{1}{1+x}H\left(\dfrac{x}{1+x}\right) = \sum\limits_{k=0}^\infty \sum\limits_{i=0}^k(-1)^{k-j}{k\choose i}x^kh_i
\end{align}
...
1
vote
1answer
31 views
Defining piecewise summation of continued fractions and rationality of sums
Let $a=[a_1,a_2\dots]$ and $b=[b_1,b_2\dots]$ be two real numbers and their continued fraction representations. They may be infinite or finite.
Let us define a thing $+^*$ so that ...
1
vote
1answer
106 views
Closed form expression for a summation over positive integers
we have the summation :
$$\underset{n\neq m}{\sum_{n=1}^{\infty}}\frac{n^{j-1}}{\left(n-m\right)^{j+1}}$$
where $j,m$ are positive integers . By partial fraction expansion, we have:
$$\underset{n\neq ...
3
votes
1answer
59 views
Show that $\frac{1}{1+x}H(\frac{x}{1+x})=\sum^\infty_{k=0}[\Delta^kh_0]x^k$
For a sequence $\{h_n\}_{\geq 0}$, let $H(x)=\sum_{n\geq0}h_nx^n$. Show that:
$$\frac{1}{1+x}H(\frac{x}{1+x})=\sum^\infty_{k=0}[\Delta^kh_0]x^k$$
What I did was that by proving the $$\Delta^k ...
1
vote
1answer
61 views
Is this sum equal to 1?
Is this function $P:\mathbb{N}\mapsto \mathbb{R}$ such that
$$
P(i)=\frac{1}{m^n}((m-i+1)^n-(m-i)^n), \quad i\in\mathbb{N}
$$
a probability over natural numbers?
I was trying to calculate if
$$
...
2
votes
1answer
61 views
How to convert from Riemann sum to integral?
Im converting this to integral: But I need help!
$$\sum_{i=1}^4{\left(-2+i\frac12\right)^3*\frac12}$$
$$\sum_{i=1}^4{\left(\frac{-13}2+\frac{47i}8\right)\frac12}$$
$$\Delta ...
1
vote
2answers
43 views
Can summations distribute across absolute values?
Can I distribute a summation as follows?
$$
k\sum_{x \in X} \left| x - b \right| = \left| \left(k\sum_{x \in X}x \right) - \left( k\sum_{x \in X}b \right) \right|
$$
0
votes
1answer
17 views
Summation sign inside an expected value
Would it be correct to assume $E\left[\sum U_i\right] = nE[U_i]$?
I am trying to show that $E[∑(U_i - E[U])^2] = (n-1)(\text{sample variance)}$.
Thanks!
0
votes
1answer
32 views
Split ${n\over2}\sum_{j\ge 1}2^{-j}(1-2^{-j})^{n-1}$ into oscillating terms.
Exercise 8.57 from Analysis of Algorithms (Sedgewick/Flajolet) asks for solving $p_n=2^{-n}\sum_k{n\choose k}p_k$ up to the oscillating term, for $p_0=0$ and $p_1=1$.
I was able to find a functional ...
1
vote
0answers
20 views
simple formula to distribute inputs
I need a simple formula to do some math for my inputs to generate max number of fixed values ..
Below i wrote a simple logic for the math
lets say we have an object that will cost fixed numbers of 4 ...
1
vote
2answers
95 views
Simplify summation of factorials
Hello I guess this equality is true but I don't know how to solve it.
$$\sum_{x=0}^{m(1-\text{sel})} (m-1-x)! (m \cdot \text{sel}) \frac{(m(1-\text{sel}))!}{(m(1-\text{sel})-x)!}(x+1) = ...
0
votes
1answer
40 views
Help with understanding a summation formula
I am having trouble understanding the derivation of the summation formula below.
$$\sum_{k=1}^N \dfrac1{(k+1)(k+2)} = \dfrac{N}{2N+4}$$
4
votes
3answers
111 views
Find a simple formula for
$$\binom{n}{0}\binom{n}{1}+\binom{n}{1}\binom{n}{2}+...+\binom{n}{n-1}\binom{n}{n}$$
All I could think of so far is to turn this expression into a sum. But that does not necessarily simplify the ...
6
votes
5answers
271 views
Simplify $\sum_{i=0}^n (i+1)\binom ni$
Simplifying this expression$$1\cdot\binom{n}{0}+ 2\cdot\binom{n}{1}+3\cdot\binom{n}{2}+ \cdots+(n+1)\cdot\binom{n}{n}= ?$$
$$\text{Hint: } \binom{n}{k}= \frac{n}{k}\cdot\binom{n-1}{k-1} $$
7
votes
3answers
170 views
Compact formula for $\sum_k k!$ [duplicate]
Is there any compact formula for:
$$\sum_{k=0}^n k!$$
I've tried to find it using one method for summation, but I was able to receive only compact formula for $\sum_k k! \cdot k = (n+1)!-1$
I've ...
0
votes
0answers
52 views
Simplification of Summation
I am trying to simplify or analysis the convergence of the following equation. $c_k$ is between $0$ and $1$. Can someone please give an idea for that?
$$ \frac{\Bigg(\sum_{k=1}^{K} ...
2
votes
3answers
45 views
Trouble summing $\frac{1}{3^i}$ from 1 to n
$\sum_{i=1}^n \frac{1}{3^i}\tag{displayed}$
I can't figure this out. I expanded it: $(\frac {1}{3^1}+\frac{1}{3^2}+...+\frac{1}{3^n})=S$, and I think the technique is to multiply both sides by ...
3
votes
1answer
43 views
Proof required that $\sum _{n=1} ^N(1-e^{(2n+1) \pi i/N})^{-1} = \frac N 2$
Numerical evidence suggests this is true, for all natural numbers $N$:
$\sum _{n=1} ^N(1-e^{(2n+1) \pi i/N})^{-1} = \frac N 2$
Can anyone prove it?
3
votes
4answers
106 views
Solve $\frac{1}{2x}+\frac{1}{2}\left(\frac{1}{2x}+\cdots\right)$
If
$$\displaystyle \frac{1}{2x}+\frac{1}{2}\left(\frac{1}{2x}+ \frac{1}{2}\left(\frac{1}{2x} +\cdots\right) \right) = y$$
then what is $x$?
I was thinking of expanding the brackets and trying to ...
3
votes
1answer
103 views
How to simplify the sum: $\sum_{m_1+…+m_L=n}\prod_{i=1}^L(t_i)^{m_i}$?
$$\sum_{m_1+ ... +m_L=n}\prod_{i=1}^L(t_i)^{m_i}$$
where $t_i$ are real numbers between $[0, 1]$ and $n$ is a positive integer. $m_i$ are integers and $0 \le m_i \le n$.
The $\sum_{m_1+...+m_L=n}$ ...
4
votes
2answers
135 views
Compact form of sum (binomial coefficients)
Find compact formula of the following sum:
$$ \sum_{i,j,k \in \Bbb Z} {{n}\choose{i+j}}{{n}\choose{j+k}}{{n}\choose{k+i}} $$
Could you give me any HINT how to start it?
I've tried this way:
$$ ...
0
votes
2answers
39 views
Summation Skip Notation
The following notation means to sum 1 to N:
$$\sum_{n=1}^N n$$
Is there a notation to not increment by one for each step, but, say, 10?
3
votes
2answers
57 views
Inequality with a sum
I am reading Remarks on a Ramsey theory for trees: Janos Pach, Jozsef Solymosi, Gabor Tardos
http://arxiv.org/abs/1107.5301
I am stuck at inequality in proof of Lemma 6.
$n\geq 8$, $k=2\lfloor ...
0
votes
0answers
37 views
Einstein notation non-repeating indices
I forget the rule for Einstein notation. If I have something like the gradient:
$$\vec\nabla f = \frac{\partial f}{\partial x_i} = \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial ...
2
votes
3answers
45 views
Sum of $\sum_{n=0}^\infty \frac{(x+2)^{n+2}}{3^n} $
Calculate the sum of the next series and for which values of $x$ it converges:
$$\sum_{n=0}^\infty \frac{(x+2)^{n+2}}{3^n}$$
I used D'Alembert and found that the limit is less than 1, so: $-5 < x ...
0
votes
1answer
57 views
Calculating a recursive power term binomial sum
Could someone please help me or give me a hint on how to calculate this sum:
$$\sum_{k=0}^n \binom{n}{k}(-1)^{n-k}(x-2(k+1))^n.$$
I have been trying for a few hours now and I start thinking it may ...
4
votes
1answer
51 views
Limit of difference of integral and sum
$f:[0,1]\rightarrow\mathbb R$ and $f\in C^1$, then the limit $\lim_{n\rightarrow\infty} n(\int_{0}^{1}f(x)dx-\frac{1}{n}\sum_{k=1}^{n}f(\frac{k-1}{n}))$ exists.
I guess the kernel lies in the sum ...
0
votes
4answers
48 views
Evaluating a summation
I am trying to solve a homework question in which a part involves the evaluation of a summation.
The summation is:
$$
\sum_{i=0}^n2^{2i+1}
$$
and this is my attempt which i am stuck at. Any lead ...
0
votes
0answers
24 views
Find the biggest sum from sequence of number which within a range
I need help. How do I find the greatest sum from sequence of number within a finite range, for example:
Given sequence {2,5,4,3,6} and the range is 11, so how to find the number within the sequence ...
0
votes
1answer
52 views
summation of fractions and inequalities
I am trying to prove that $\sum_{i=1}^{n}\frac{1}{a_i}\leqslant 2$, where all $a_i$ are less than 1000, and all $a_i$ have a lowest common multiple greater than 1000.
This is what I have done so far:
...
5
votes
2answers
152 views
use residues to evaluate sum involving square of csch
I have been trying to evaluate the following sum using residues
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{\sinh^{2}(\pi n)}=\frac{1}{6}-\frac{1}{2\pi}$
I am mainly interested in using residues to ...
1
vote
2answers
38 views
How is this sum calculated?
We have $N$ letters to $N$ different people, and $N$ envelopes addressed to those $N$ people. One letter is put in each envelope at random. Find the mean and variance of the number of letters ...
1
vote
1answer
91 views
Sum of the first k binomial coefficients for fixed n
I am reading Remarks on a Ramsey theory for trees by Janos Pach, Jozsef Solymosi and Gabor Tardos.
Let $k, d, n \geq 2$ be integers. Somethig interesting happens when $$2^{n/k} > \sum_{i=0}^{d-1} ...
2
votes
2answers
76 views
How to turn this into an equation and then sum the series?
I'm reading "How would you move Mount Fuji?", and one of the puzzzles/questions is:
A train leaves Los Angeles for New York at a constant
speed of 15 miles an hour. At the same moment, a train
...
