Tagged Questions
-2
votes
1answer
61 views
I asked a question related to it, but are my findings meaningless?
I asked a question related to it and found something interesting (at least that is what I think)...
I went through a sequence in which I had to find the wrong term and now I think the correct ...
4
votes
2answers
62 views
What does $\lim \limits_{n\rightarrow \infty }\sum \limits_{k=0}^{n} {n \choose k}^{-1}$ converge to (if it converges)? [duplicate]
How we can show if the sum of $$\lim_{n\rightarrow \infty }\sum_{k=0}^{n} \frac{1}{{n \choose k}}$$ converges and then find the result of the sum if it converges?
Thanks for any help.
0
votes
1answer
28 views
Inverting an infinite sequence transformation
Consider a sequence $\{b_k\}$ define via:
$$
b_k = \sum_{n=0}^\infty \frac{(n+k)!}{n!}a_n.
$$
I would like to invert this transform. That is, I would like to know the coefficients $c_{nk}$ such that
...
1
vote
2answers
65 views
Binomial Expansion Word Problem (Creating a Equation)
I was working on my math textbook (Nelson Functions 11) and came across the following word problem. This question is shown in the "Pascal's Triangle and Binomial Expansions" section of the book.
...
8
votes
1answer
152 views
Proof of $\sum_{n=1}^\infty \frac{1}{n^4 \binom{2n}{n}}=\frac{17\pi^4}{3240}$
Recently, I was able to prove that
$$\sum_{n=1}^\infty \frac{1}{n \binom{2n}{n}}= \frac{\pi}{3\sqrt{3}}$$
$$\sum_{n=1}^\infty \frac{1}{n^2 \binom{2n}{n}}= \frac{\pi^2}{18}$$
But does anybody know ...
2
votes
4answers
114 views
Is $\sum\limits_{k=1}^{n-1}\binom{n}{k}x^{n-k}y^k$ always even?
Is
$$ f(n,x,y)=\sum^{n-1}_{k=1}{n\choose k}x^{n-k}y^k,\qquad\qquad\forall~n>0~\text{and}~x,y\in\mathbb{Z}$$
always divisible by $2$?
0
votes
1answer
39 views
What is the function given by $\sum_{n=0}^\infty \binom{b+2n}{b+n} x^n$, where $b\ge 0$, $|x| <1$
For a nonnegative integer $b$, and $|x|<1$, what is the function given by the power series
$$
\sum_{n=0}^\infty \binom{b+2n}{b+n} x^n.
$$
For $b=0$, this post shows
$$
\sum_{n=0}^\infty ...
1
vote
3answers
34 views
Proof that $\sum_{k=0}^m \binom{m}{k}\frac{1}{k+1} = \frac{2^{m+1}-1}{m+1}$ [duplicate]
Recently I needed to compute $E[\frac{1}{X+1}]$ where $X\sim Bin(m, \frac 1 2)$.
While expanding, I came across the sum $\sum_{k=0}^m \binom{m}{k}\frac{1}{k+1}$, which I was unable to solve. Plugging ...
1
vote
3answers
139 views
Evaluate a sum with binomial coefficients
$$\text{Find} \ \ \sum_{k=0}^{n} (-1)^k k \binom{n}{k}^2$$
I expanded the binomial coefficients within the sum and got $$\binom{n}{0}^2 + \binom{n}{1}^2 + \binom{n}{2}^2 + \dots + \binom{n}{n}^2$$
...
2
votes
2answers
38 views
Prove that a sum converges to a trigonometric expression
$$2^n \cos \left (\frac{n \pi}{2} \right )=\sum_{k=0}^{n} (-1)^k \binom{2n}{2k}$$
I expanded the LHS and got $$\binom{2n}{0}-\binom{2n}{2}+\binom{2n}{4}-\binom{2n}{6}+\cdots+(-1)^{n}\binom{2n}{2n}$$
...
0
votes
0answers
27 views
Sum involving binomial cofficients [duplicate]
I want to Solve a binomial Series of type :
aC0*bCd + aC1*bc(d-1) -----------------(aC(k-1))*(bCd-(k-1))
Can anyone please suggest on how to reduce such series?
...
4
votes
0answers
65 views
Sum with binomial coefficients and a square root
I encountered this sum from working on an integral:
$$\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\sqrt{k}$$
I don't think it can be written as a hypergeometric function, because of this square root.
Does ...
5
votes
5answers
129 views
Closed-form expression for $\sum_{k=0}^n\binom{n}kk^p$ for integers $n,\,p$
Is there a closed-form expression for the sum $\sum_{k=0}^n\binom{n}kk^p$ given positive integers $n,\,p$? Earlier I thought of this series but failed to figure out a closed-form expression in $n,\,p$ ...
3
votes
0answers
33 views
Proving two summations equivalent [duplicate]
Let $h_n$ be an infinite sequence. I need to show that:
\begin{align}\dfrac{1}{1+x}H\left(\dfrac{x}{1+x}\right) = \sum\limits_{k=0}^\infty \sum\limits_{i=0}^k(-1)^{k-j}{k\choose i}x^kh_i
\end{align}
...
1
vote
1answer
36 views
Sum of $nC^k$ and $k*nC^k$
How to find
$$\sum_{k=0}^n nC^k$$
and
$$\sum_{k=0}^n knC^k$$
Does this help : $\sum n=\frac{n(n+1)}{2}?$
1
vote
2answers
171 views
Prove that $\sum\limits_{i=1}^n 2i\binom{2n}{n-i}= n\binom{2n}{n}$
Let n be a positive integer.
Prove that $$\sum_{i=1}^n 2i\binom{2n}{n-i}= n\binom{2n}{n}$$
11
votes
3answers
259 views
Closed form for $\sum_{k=0}^{n} k\binom{n}{k}\log\binom{n}{k}$
Is it possible to write this in closed form:
$$\sum_{k=0}^{n} k\binom{n}{k}\log\binom{n}{k}$$
Can you get something like $$n2^{n-1}\log(2^{n-1})$$
0
votes
1answer
88 views
Using the general binomial theorem to find a series-like expression for $\sqrt 2$
How do I use the general binomial theorem (i.e. the series expansion of ${(1+x)^\alpha}$ for $ |x|<1$) to show the following? $$\sqrt 2=1+\frac 1{2^2}+\frac{1\cdot3}{2!\cdot{2^4}} ...
3
votes
2answers
75 views
How to get the sum of the values in a $N \times N$ table?
How to get the sum of the values in a $N \times N$ table (without adding repeating products such as $6 \times 7$ and $7 \times 6$ twice and without counting perfect squares)?
Figured out that
$1 ...
4
votes
2answers
112 views
Is there a closed form expression for the first half of the Binomial series?
I'm looking for a closed form expression for the sum
$P_n(x) =\sum_{0\leq k\leq n/2}\binom{n}{k}x^k$,
where $n$ is a given positive integer and $k$ runs over nonnegative integers between $0$ and ...
7
votes
1answer
200 views
Summation of an Infinite Series: $\sum_{n=1}^\infty \frac{4^{2n}}{n^3 \binom{2n}{n}^2} = 8\pi G-14\zeta(3)$
I am having trouble proving that
$$\sum_{n=1}^\infty \frac{4^{2n}}{n^3 \binom{2n}{n}^2} = 8\pi G-14\zeta(3)$$
I know that
$$\frac{2x \ \arcsin(x)}{\sqrt{1-x^2}} = \sum_{n=1}^\infty ...
2
votes
2answers
99 views
Finding Binomial expansion of a radical
I am having trouble finding the correct binomial expansion for $\dfrac{1}{\sqrt{1-4x}}$:
Simplifying the radical I get: $(1-4x)^{-\frac{1}{2}}$
Now I want to find ${n\choose k} = ...
0
votes
1answer
144 views
Sum of following binomial series :
I need to solve this binomial summation but cant seem to get it using binomial identities I learnt in school and college first-year:
$$S=\sum_{i=q}^{p-q}{\binom{i}{q}}{\binom{n-i}{p-q}}$$
p,q,n are ...
12
votes
1answer
247 views
Binomial sum of $n$ terms in closed form
Can we calculate the given combinatorial sum in closed form?
$$ \frac{\binom{2}{0}}{1}+\frac{\binom{4}{1}}{2}+\frac{\binom{8}{2}}{3}+\frac{\binom{16}{3}}{4}+\cdots+\frac{\binom{2^n}{n-1}}{n}$$
5
votes
1answer
179 views
Evaluate $\sum_{n=1}^{\infty} \frac{(2n)!}{2^{2n}(n!)^2 (2n-1)}$
How to evaluate the series
$$S = \sum_{n=1}^{\infty} \frac{1}{2^{2n}(2n-1)} \binom{2n}{n}$$
The original question was to show that for
$$ a_n=\left(\frac{ 2n-3 }{ 2n }\right)a_{n-1} , a_1 = \frac 1 ...
4
votes
4answers
167 views
Closed form for $\sum_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{(k+1)^2}$
How can I calculate the following sum involving binomial terms:
$$\sum_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{(k+1)^2}$$
Where the value of n can get very big (thus calculating the binomial ...
1
vote
2answers
38 views
Proof that $\lim_{n\rightarrow\infty}\frac{n^\alpha}{(1+p)^n}=0$ from Rudin's Principles of Mathematical Analysis, involving the binomial theorem
We are showing that when $\alpha$ and $p$ are real and $p>0$ then $$\lim_{n\rightarrow\infty}\frac{n^\alpha}{(1+p)^n}=0$$
Proof. Let $k$ be an integer such that $k>0$, $k>\alpha$. Then for ...
14
votes
5answers
1k views
Crazy induction
I found crazy (for me at least) induction example, in fact it just would be nice to prove. (Even have problems with starting) Any hints are highly valued: ...
0
votes
3answers
91 views
Binomial sum of a sequence
I have the following sequence:
$$
\sum\limits_{k=1}^n k\binom{n-1}{k-1}
$$
What is the sum of this sequence.
6
votes
3answers
160 views
What is $\lim\limits_{n\to\infty} \frac{n^d}{ {n+d \choose d} }$?
What is $\lim\limits_{n\to\infty} \frac{n^d}{ {n+d \choose d} }$ in terms of $d$? Does the limit exist? Is there a simple upper bound interms of $d$?
8
votes
2answers
204 views
Techniques for summing ratio of binomial coefficients
There are several identities that involve the sum of the product of binomial coefficients. However what I am searching for is an identity that involves the ratio of binomial coefficients. ...
3
votes
2answers
168 views
what is the easiest way to represent $ \sqrt{1 + x} $ in series
How to expand $ \sqrt{1 + x}$.
$$ \sum_{n = 0}^\infty {{\left ( 1 \over 2\right )!}x^n \over n! \left({1 \over 2 }- n\right )!} = 1 + \sum_{n = 1}^\infty {{\left ( 1 \over 2\right )!}x^n \over n! ...
15
votes
3answers
581 views
Alternating sum of squares of binomial coefficients
I know that the sum of squares of binomial coefficients is just ${2n}\choose{n}$ but what is the closed expression for the sum ${n}\choose{0}$$^2$ - ${n}\choose{1}$$^2$ + ${n}\choose{2}$$^2$ + ... + ...
2
votes
1answer
214 views
proof of a finite sum involving a binomial coefficient and a variable.
I found that the following equation holds for integers $l$, $k$, and any $x \neq 0,1$,
$$\tag{1}
\sum\limits_{l = 0}^k {\left( { - 1} \right)^l } \left( {\begin{array}{*{20}c}
k \\
l \\
...
1
vote
2answers
294 views
modified $\sum{k{n \choose k}}$ closed form expression
There is probably something stupidly simple I'm missing, but I'm trying to find a closed form for:
$$
2\sum_{k=1}^{(n-1)/2} k \, {n \choose k} \hspace{1cm} (n\textrm{ is odd})
$$
Anyone know how to ...
8
votes
1answer
116 views
A three variable binomial coefficient identity
I found the following problem while working through Richard Stanley's Bijective Proof Problems (Page 5, Problem 16). It asks for a combinatorial proof of the following:
$$ \sum_{i+j+k=n} ...
1
vote
4answers
238 views
Alternating sum of binomial coefficients
Calculate the sum:
$$ \sum_{k=0}^n (-1)^k {n+1\choose k+1} $$
I don't know if I'm so tired or what, but I can't calculate this sum. The result is supposed to be $1$ but I always get ...
2
votes
2answers
254 views
Does this qualify as a proof? (Spivak's 'Calculus')
I'm working through Spivak's 'Calculus' at the moment, and a question about series confused me a bit. I think I have the solution, but I'm not sure if my "proof" holds.
The question is:
Prove ...
11
votes
1answer
201 views
A binomial identity
I was wandering if someone knows an elementary proof of the following identity:
$$
\frac{(a)_n (b)_n}{(n!)^2} = \sum_{k=0}^n (-1)^k {1-a-b \choose k}
\frac{(1-a)_{n-k}(1-b)_{n-k}}{((n-k)!)^2}\ ,
$$
...
3
votes
3answers
164 views
Computing a sum of binomial coefficients
Does anyone know a better expression than the current one for this sum?
$$
\sum_{i=0}^m \binom{N-i}{m-i}, \quad 0 \le m \le N.
$$
It would help me compute a lot of things and make equations a lot ...
4
votes
0answers
88 views
How to transform series of series into series
I need to prove this equation.
$$ \sum_{k=0}^{i-2} \left( e \space α(k+1)\space\frac{(-1)^{i+k+2}}{(i-k-2)!} \right) = \sum_{k=0}^{i-2} \frac{(i-k)^k}{k!} \space e^{i-k} (-1)^k\space where,\space α(i) ...
6
votes
2answers
162 views
Find the sum of this series :$ \frac{1}{{1!2009!}} + \frac{1}{{3!2007!}} + \cdots + \frac{1}{{1!2009!}}$
Find the sum of this series :
$$\sum\limits_{\scriptstyle 1 \leqslant x \leqslant 2009 \atop
{\scriptstyle x+y=2010 \atop
\scriptstyle {\text{ }}x,y{\text{ odd}} }} {\frac{1}{{x!y!}}} = ...
3
votes
1answer
314 views
Multiplication of Two Infinite Series
This question has been deleted.
How to prove that
$$\displaystyle \left( \sum_{k=0}^{\infty }\frac{\left( -a\right) ^{k}y^{2k}}{k!}\right)
\left( \sum_{k=0}^{\infty }\frac{a^{k}y^{2k+1}}{\left( ...
1
vote
1answer
169 views
Using binomial theorem find general formula for the coefficients
Using binomial thaorem (http://en.wikipedia.org/wiki/Binomial_theorem) find the general formula for the coefficients of the expantion:
$$
...
1
vote
0answers
62 views
calculation of the sum using idea of one answer
I am wondering if the sum (the $q$-th moment) in my question Calculation of the moments using Hypergeometric distribution can be calculated using idea in Evaluating 'combinatorial' sum ?
...
0
votes
2answers
101 views
Help with $1 + a + a(a-1) + a(a-1) (a-2) +\cdots+a(a-1)\cdots(a-(n-1))$
I want to rewrite the series $$1 + a + a(a-1) + a(a-1) (a-2) +\cdots+a(a-1)\cdots(a-(n-1))$$ as $(a^n-1)Y$ or $(a^{n-1}-1)Y$
Short-form:
$$\{1+\sum_{i=1}^{n} \prod_{j=0}^{i-1}(a-j)\}$$
as $(a^n-1)Y$ ...
2
votes
1answer
254 views
Evaluating 'combinatorial' sum
Help me please to calculate the following sum. I have seen such kind of formulas in the papers related to combinatorics, specifically 'trees'. I am curious how to calculate or approximate this sum:
...
4
votes
1answer
113 views
Calculate $\sum_{i=1}^{[\frac{\sqrt n}{2}]}{n\choose i}$
It is known that $\sum_{i=1}^n {n \choose i}=2^n$. I am wondering what would be the sum if we change the upper limit to $\sqrt n/2$, i. e. How to calculate$$\sum_{i=1}^{[\frac{\sqrt n}{2}]}{n \choose ...
0
votes
0answers
108 views
Calculation of sum
I am wondering if it is possible to calculate or approximate the following sum
$$
\sum_{k=0}^l\frac{(l-2k)^p(2l+k(k-1))l^{k-1}}{(k+3)(k+2)}
$$here $p \geq 2$.
Thank you.
0
votes
1answer
92 views
Identity of binomial coefficients with a series
I never really used any series/infinite sums and now I should proove the following identity:
$$\sum\limits_{k=0}^{\infty}\binom{m}{k}\binom{n}{l-k}=\binom{m+n}{l}$$
Can you please explain me, how to ...
