How can I prove this series?

$$\frac{x}{\sqrt{(x+1)}} = \frac{x}{x+1}+ \frac{x^2}{2(x+1))^{2}} + \frac{3x^3}{4(x+1))^{3}} +\dots$$

I tried using Maclaurin's series.But obviously I don't get this form as the series is not just in terms of powers of $x$ but $\dfrac{x}{x+1}$.

I also tried to find the $S_{inf}$ (sum) for the right hand series but I couldn't get a closed form. It's not a homework problem.I just saw this problem in a book and I thought I could solve this but I couldn't.

• For the sake of clarity, please write the general term of the RHS series. – mathcounterexamples.net Nov 7 '15 at 15:34
• can you give us the reference of this series? – E.H.E Nov 7 '15 at 16:05
• It's from a book and the book is not in English and I want to point something the series is x/(x+1) + 1/2 (x/(x+1))^2 + 3/4(x/(x+1))^3 .. Someone edited it wrong – anon Nov 7 '15 at 16:39

Writing $$t:=\frac{x}{x+1}$$ makes the function at interest
$$\sqrt{x} \sqrt{t}=\sqrt{\frac{t}{1-t}}\sqrt{t}=\frac{t}{\sqrt{1-t}}.$$
Now the problem is reduced to expressing the above in powers of $t$, which is done through the Maclaurin series around $t=0$.