5
votes
3answers
569 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?
0
votes
5answers
225 views

Proof of a binomial identity $\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$

Prove that $$\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$$ The exercise provides the following hint: $\,\,\displaystyle{n \choose k}={n\choose n-k}$. Any help?
1
vote
0answers
86 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 = ...
1
vote
2answers
39 views

How to evaluate binomial coefficients when $k=0$ and $1\geq|n|\geq0$

So I am doing some taylor series which boil down to the geometric series, for which I then need to evaluate various binomial coefficients. I have always used $n\choose k$ $=\frac{n!}{(n-k)!k!}$ to do ...
1
vote
0answers
63 views

Binomial Expansion problem error

I tried solving this question but failed. a) Expand $(1+2x)^{1/4}$ in ascending powers of $x$ up to and including the term in $x^3$, simplifying each term as far as possible. b) By substituting ...
1
vote
6answers
224 views

How to simplify this expression by division

How to divide it $${\frac {{x}^{n-2}-{y}^{n-2}}{x-y}}$$ to remove the $x-y$ term from the denominator. We may assume that $n>2$ is an integer. Thanks.
1
vote
1answer
370 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: $$ ...
6
votes
1answer
1k views

Using the Taylor expansion for ${(1+x)}^{-1/2}$, evaluate $\sum_{n=0}^\infty \binom{2n}{n} a^n$

Using the Taylor expansion for $${(1+x)}^{-1/2}$$ we have $${(1+x)}^{-1/2}= \sum_{n=0}^\infty \binom{-1/2}{n} (x^n)$$ for $|x|<1$. But if $|a| <1$, how can we use the above fact to find ...