2
votes
1answer
43 views

Infinite series containing binomial coefficients

I've encountered the following series: $$\sum_{t=1}^\infty {1 \over 2^{t}}\, {{\large t} \choose {\large{t + x \over 2}}}$$ Is this series even convergent? I'm really lacking knowledge on series ...
0
votes
1answer
41 views

Binomial coefficient - first two terms, proof of inequality

I've seen the following and I'm not sure whether it is true or not, and if yes, why it holds. $(1-p)^x \geq 1-p x$ for $p\in (0,1)$ and $x>0$. Do I need some additional Information to prove ...
7
votes
2answers
94 views

Does this really converge to 1/e? (Massaging a sum)

Short version: can we prove that $$\sum_{k=0}^n (-1)^k \binom{n}{k}^2 \frac{k!}{n^{2k}} \to \frac1e$$ as $n \to \infty$? Long version: First, consider $$a_n = \sum_{k=0}^n \frac{(-1)^k}{k!}$$ It is ...
2
votes
0answers
33 views

Sum of Binomials times Logarithms

Is there a closed-form expression or a very good approximation for $$ \sum_{i=0}^n \binom{n}{i} \log (i+1) \,? $$ If the summands alternate, then there is a very close approximation, yet it feels ...
3
votes
3answers
98 views

Binomial theorem $(a+b)^n=\sum \limits_{k=0}^{n}\binom{n}{k}a^{n-k}b^{k}$ [duplicate]

I'm trying to understand the proof by induction of: $$ (a+b)^n = \sum \limits_{k=0}^{n}\binom{n}{k}a^{n-k}b^{k} $$ I'm at the point of deriving the inductive step and am getting next: $$ (a+b)^{n+1} = ...
5
votes
6answers
218 views

Asymptotic behavior of $\sum_{k=1}^n \binom{n}{k} \left(\frac{ck}{n}\right)^k$

I am looking to show that $$\lim_{n \rightarrow \infty}\frac{1}{e^n}\sum_{k=1}^n \binom{n}{k} \left(\frac{ck}{n}\right)^k = 0. $$ In my application, $c = (e+1)/2 \approx 1.85914\ldots$. I have been ...
2
votes
2answers
76 views

Infinite Sum with Combination

I am trying to figure out what the following sum converges to: $$\sum_{n=0}^\infty {6+n\choose n}x^n(6+n),\qquad\qquad0<x<1$$ An answer would be great, but if you have an explanation, that'd ...
5
votes
2answers
107 views

Prove that $\sqrt{8}=1+\dfrac34+\dfrac{3\cdot5}{4\cdot8}+\dfrac{3\cdot5\cdot7}{4\cdot8\cdot12}+\ldots$

Prove that $\sqrt{8}=1+\dfrac34+\dfrac{3\cdot5}{4\cdot8}+\dfrac{3\cdot5\cdot7}{4\cdot8\cdot12}+\ldots$ My work: $\sqrt8=\bigg(1-\dfrac12\bigg)^{-\frac32}$ Now, I suppose there is some "binomial ...
10
votes
3answers
287 views

Prove that $\,\,\displaystyle\inf_{n\in\mathbb N}\sum_{k=0}^{p}\lvert\sin{(n+k)^p}\rvert>0$

For any poistive integer number $p$, show that $$\inf\left\{ {\left\vert\sin{(n^p)}\right\vert+\left\vert\sin{(n+1)^p}\right\vert+\cdots+ \left\vert\,\sin{(n+p)^p}\right\vert\, :\,n\in ...
1
vote
0answers
45 views

Increasing fraction of sum of binomial coeffients

Let $n$ be a positive integer. Show that the quantity $$ \displaystyle \frac{ \displaystyle \sum_{i=1}^n { n+k \choose i-1 } }{ \displaystyle \sum_{i=1}^n { n+k+1 \choose i } } $$ is ...
4
votes
1answer
110 views

Recursive sequence with binomial coefficients

I have a sequence $\epsilon_i$ defined recursively for $i\ge 1$ as follows \begin{eqnarray*} \epsilon_1 &=& \frac{1}{p}\\ \epsilon_n &=& \frac{1}{1-(1-p)^n}\left( 1 + \sum_{j=1}^{n-1} ...
4
votes
2answers
76 views

Series of inverses of binomial coefficients

Can you think of a simple way of proving that $$ \sum_{n=k+1}^\infty \frac{1}{n \choose k} $$ is rational for any $k \geq 2$? Here's the background. Consider a series: $$ \sum_{n=1}^\infty ...
0
votes
2answers
98 views

Upper bound for $\sum_{j=0}^i {i \choose j}^{n}$

Is there an upper bound for sums of powers of binomial coefficients? I have $$\sum_{j=0}^i {i \choose j}^{n}$$ where $n$ is a positive integer. I am hoping this will help me solve Limit for a ...
16
votes
4answers
515 views

Limit of $\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1}\right)$

I'm trying to calculate the limit for the sum of binomial coefficients: $$S_{n}=\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1} \right).$$
0
votes
1answer
22 views

Validating a proposition

Proposition: For all $k,n\in\mathbb{Z^+}$ $s.t$ $n\lt4$ $2{n\choose n}+{n\choose n-1}+...+{n\choose k-(n-2)}=2^n$ for $1\le k\le n-1.$ I understand that this proposition is invalid, so are there ...
2
votes
2answers
44 views

Upper bound for sum

I am trying to get an upper bound the following sum: $$S_{n,r}=\sum_{i=0}^n \binom{n}{i} \left(\frac{\binom{n}{i}}{2^n}\right)^{r} .$$ Any hints would be greatly appreciated. I thought of using ...
1
vote
1answer
50 views

Express number of ways integer can be written as coefficient in generating series

Question: "Express the number of ways that an integer $n$ can be written as a sum of a cube of an integer $s\ge-1$ plus the fourth power of an integer $t$ plus the square of an odd integer $r$ as a ...
10
votes
2answers
107 views

Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $\sum_{k=1}^n\frac{a_k-b_k}k$

Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $$\sum_{k=1}^n\frac{a_k-b_k}k$$ By computing some partial sums, the answers are 0. It seems an inductive argument is possible.
2
votes
1answer
80 views

Is anyway to prove this: $\prod_{k=1}^{n}(a_{k})< (1/n^n)*(\sum_{k=1}^{n}(\sqrt{1+a_{k}*a_{k+1}}))^n$

$$ \prod\limits_{k=1}^{n}a_{k} < {1 \over n^{n}}\left(\,\sum_{k = 1}^{n}\,\sqrt{1+a_{k}\,a_{k+1}\,}\,\right)^n $$ ak and n are positive real number greater than 0. EDIT: a_{k+1} becomes a_{1} ...
2
votes
1answer
90 views

How to evaluate $\sum\limits_{k=0}^{n} \sqrt{\binom{n}{k}} $

Can we find $$ \sum_{k=0}^{n} \sqrt{\binom{n}{k}} \quad$$ This problem asked me my friend about a year ago, but I didn't know how to attack problem. Now, I am interesting in solution. Any suggestion? ...
0
votes
0answers
43 views

Does $\sum_{k=0}^n {n \choose k} x^{k^2}$ have a closed form?

Does $\sum_{k=0}^n {n \choose k} x^{k^2}$ have a closed form, similar to $\sum_{k=0}^n {n \choose k} x^{k} = (1+x)^n$?
4
votes
1answer
85 views

Simplify $\sum_{k=0}^n \frac{1}{k!(n-k!)}.$

Is there a way to simplify the expression $$\sum_{k=0}^n \frac{1}{k!(n-k)!}?$$ This came up when I was trying to determine $\mathbb{P}(X+Y =r)$ given a joint mass probability $$m_{X,Y}(j,k) = ...
2
votes
0answers
115 views

Getting K heads out of N biased coins problem (formula generation ).

Problem- Given a set of coins n with each coin i having Pi probability to give heads. Find the probability of getting k heads, when all coins are tossed together. hi i have solved this problem ...
0
votes
1answer
46 views

How to expense $(a+b)^\alpha$ into multinomial with $\alpha \in \mathbb{R}$?

As we all know, the binomial expension is as follows $$ (a+b)^2 = a^2 +2ab +b^2. $$ When the power number is a real number, not a integral, how to expense $(a+b)^\alpha$ into multinomial with $\alpha ...
1
vote
2answers
68 views

Closed form of the limit of a sequence (weighted average)

I have a sequence, which can actually be seen as Riemann-Stieltjes integration with a binomial distribution. $\rho \in (0,1)$. $$ S_N ...
1
vote
1answer
76 views

Prove this conjecture

I come across an equality to complete a proof in my paper. I think it is true and I confirm by numerically experimenting with different parameter values, and analytically proving this with n=1,2. But ...
20
votes
1answer
404 views

Evaluating $\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} $

Wolfram MathWorld states that $$ \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = \frac{ \pi \sqrt{3}}{18} \Big[ \psi_{1} \left(\frac{1}{3} \right) - \psi_{1} \left(\frac{2}{3} \right) \Big]- ...
0
votes
2answers
61 views

Binomial and series with 2 coefficients

I would be very grateful if you would help me with this question: Find the sum : $$ \sum_{k=0}^{n}\binom{2n}{k} $$
1
vote
0answers
81 views

$\sum_{k=0}^{\infty}\frac1{4^k(2k+1)}\binom{2k}{k}x^{2k+1}\sum_{k=0}^{\infty}(-1)^k\frac1{4^k}\binom{2k}{k}x^k =$

Prove that for $|x|<1$, $\sum_{k=0}^{\infty}\frac1{4^k(2k+1)}\binom{2k}{k}x^{2k+1}\sum_{k=0}^{\infty}(-1)^k\frac1{4^k}\binom{2k}{k}(-x^2)^k = ...
10
votes
4answers
344 views

Show that if $\prod\limits_{k=1}^{n}(x+a_k)=\sum\limits_{k=0}^{n} {n\choose k}a^k_kx^{n-k}$ then $a_1=a_2=a_3=…=a_{n-1}=a_n$

Let $a_0=1$. Prove that, if $$\prod_{k=1}^{n}(x+a_k)=\sum_{k=0}^{n} {n\choose k}a^k_kx^{n-k}=x^n+{n\choose 1}a_1x^{n-1}+{n\choose 2}a^2_2x^{n-2}+....+a^n_n,$$ then ...
8
votes
3answers
244 views

Sum of squares of binomial coefficients

I came across the following sum in reference to this question $$\sum_{n=0}^{\infty} \frac{1}{2^{5 n}} \binom{2 n}{n}^2 = \frac{\sqrt{\pi}}{\Gamma \left( \frac{3}{4}\right)^2}$$ The sum on the left ...
4
votes
2answers
212 views

How find the value $\sum\limits_{k=1}^{\infty}(C_{3k}^k)^{-1}$

find the value $$\sum_{k=1}^{\infty}\dfrac{1}{C_{3k}^{k}}$$ and long ago,I have see this and is easy $$\sum_{k=1}^{\infty}\dfrac{1}{C_{2k}^{k}}$$ where $$C_{n}^{k}=\dfrac{n!}{(n-k)!k!}$$
-2
votes
1answer
98 views

Is this just a version of the binomial theorem?

I asked a question related to it and found something interesting (at least that is what I think)... Here is the link to the original question: What is the pattern of this sequence? I went through a ...
4
votes
2answers
105 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.
1
vote
1answer
59 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
918 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. ...
12
votes
1answer
281 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
172 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
40 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
51 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
2answers
351 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
56 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
29 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? ...
6
votes
0answers
139 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 ...
6
votes
5answers
242 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
35 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
120 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
235 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
281 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
166 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}} ...