Tag Archives: central binomial numbers

am I a little mathemagizian?


Well, let me start by saying that some kind of maths are my bizzness and the things that  i like to delve deep into: topology and algebra.

I like series like 

\sum_{n=1}^{\infty}\frac{2 x^{n-1}}{n(n+1)}=1+\frac{x}{3}+\frac{x^2}{6}+\frac{x^3}{10}+\frac{x^4}{15}+\cdots

\sum_{n=0}^{\infty}\frac{x^n}{{2n\choose n}}=1+\frac{x}{2}+\frac{x^2}{6}+\frac{x^3}{20}+\frac{x^4}{70}+\cdots 

or 

 \sum_{n=0}^{\infty}\frac{(n+1)x^n}{{2n\choose n}}=1+x+\frac{x^2}{2}+\frac{x^3}{5}+\frac{x^4}{14}+\cdots

Also, let me ask you: do you know how these three series are related or how are their generating functions?

 

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