Finding the sum $\frac{1}{3\times5}+\frac{1}{5\times7}+\frac{1}{7\times9}+\cdots$ The sum of series is
$${1\over(3\times5)}+{1\over(5\times7)}+{1\over(7\times9)}+\cdots$$
I used to solve this problem as its $n$th term
$${1\over (2n+1)(2n+3)}$$
Now how can I proceed??

Let there are functions $$f(x) = \sum_{n=1}^\infty \frac{x^n}{\alpha n + \beta}$$ 
and $$\psi(x) = \frac{f(x,\alpha,\beta) - f(x,\alpha,\alpha + \beta)}{\alpha} = \sum_{n=1}^\infty \frac{x^n}{(\alpha n + \beta)(\alpha (n+1) + \beta)}$$
where $\alpha,\beta > 0$ and $x \in [-1,1]$.
Thereby $$\psi(x) = \frac{1}{\alpha}\left(\frac{x}{\alpha + \beta} + \lim_{n\to \infty} \frac{x^n}{\alpha(n+1) + \beta}\right) = \frac{x}{\alpha(\alpha + \beta)} .$$
For instance that $\alpha = 2,\beta = 1$ and $x = 1$, $$\psi(1) = \frac{1}{6}.$$
