5
votes
3answers
594 views

Why does this infinite series equal one?

Why does $$\sum_{k=1}^\infty \binom{2k}{k} \frac{1}{4^k(k+1)}=1$$ Is there an intuitive method by which to derive this equality?
3
votes
2answers
83 views

Prove by induction that $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ is decreasing

I want to prove that the following sequence is monotonously decreasing: $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ I think it should be ...
3
votes
1answer
125 views

How to prove that $\sum_{k=0}^\infty \binom{x}{x-k}\cdot\binom{x}{k-x} = 1$?

How to prove this: $$\sum_{k=0}^\infty \binom{x}{x-k}\cdot\binom{x}{k-x} = 1$$ For all $x\in\mathbb R_{\ge0}$ and with $\binom{x}{r}=\frac{\Gamma(x+1)}{\Gamma(r+1)\cdot\Gamma(x-r+1)}$ It is obviously ...
2
votes
1answer
47 views

Identity with binomials [duplicate]

Does there exist a closed formula for $$\underset{n=1}{\overset{N-1}{\sum}}\dbinom{N+n}{n}?$$ I've searching on wikipedia but I haven't found this kind of sum.
5
votes
2answers
131 views

Evaluating a sum involving binomial coefficient in denominator

I came across the following sum: $$\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}\frac{4^k}{{2k \choose k}}$$ I thought that this can be evaluated using the expansion of ...
1
vote
1answer
33 views

How prove $\sum_{k=0}^{2m}\binom{2m+1}{k}\cdot 2^k\cdot B_{k}=0$

we know Bernoulli number such identity $$\sum_{k=0}^{n}\binom{n+1}{k}B_{k}=0$$ see:Bernoulli number identity show that $$\sum_{k=0}^{2m}\binom{2m+1}{k}\cdot 2^k\cdot B_{k}=0$$ where $B_{n}$ ...
4
votes
2answers
187 views

Generating function of binomial coefficients ${n\choose5}$

How to prove easily this identity for (almost classical) series with binomial coefficients: $$ \sum_{n=5}^\infty \dfrac{\binom{n}{5}}{2^{n+1}} = 1 . $$ Thank you. Any smart proof would be much ...
9
votes
3answers
299 views

Help with combinatorial proof of identity: $\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$

How to prove this identity? Can someone please give me some insight ? $$\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$$
1
vote
1answer
35 views

Help with this hard recurrence relation question

Please help with this. Suppose $\{a_n\}$ satisfies $$a_n=(n+1)a_{n-1}-(n-2)a_{n-2}-(n-5)a_{n-3}+(n-3)a_{n-4},$$ and $a_0=a_1=1,a_2=a_3=0$. Please sort out the general form of $a_n$. I guess $a_n$ ...
1
vote
0answers
49 views

Approximation of the following mathematical formula

I have the following mathematical expression which I need to simplify: $$\mu^2\sum_{x=0}^{n}\left(\frac{\theta}{\mu}\right)^x\frac{1}{H_x}{n+a\choose x}$$ $\mu$, $\theta$, $D$, and $a$ are ...
2
votes
1answer
58 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
46 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
128 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
37 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
116 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
239 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
111 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
109 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 ...
11
votes
3answers
307 views

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

For any positive 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
54 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
120 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
91 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
100 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
517 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
51 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
57 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
118 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
83 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} ...
0
votes
0answers
54 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
89 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
157 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
48 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
70 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
81 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
443 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
64 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
87 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
382 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
2answers
308 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
218 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
101 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
110 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
66 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
1k 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
331 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
181 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
43 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
56 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
380 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 $$\binom{n}{0}^2 + \binom{n}{1}^2 + \binom{n}{2}^2 + \dots + \binom{n}{n}^2$$ ...