Prove $\sum \frac{t}{(1+y)^t }= \frac{y+1}{y^2}$

Question

I see on Wolfram Alpha that $\sum \frac{t}{(1+y)^t} = \frac{y+1}{y^2}$ when t goes to infinity. I cannot, however, proove it myself. What theory is used and how do I start the proof?

Answer 1

Recall from Geometric series, we have $$\sum_{t=0}^{\infty} x^t = \dfrac1{1-x}$$ Differentiating both sides, we obtain $$\sum_{t=0}^{\infty} tx^{t-1} = \dfrac1{(1-x)^2} \implies \sum_{t=0}^{\infty} tx^{t} = \dfrac{x}{(1-x)^2}$$ Plugging in $x=\frac1{1+y}$, we obtain what you want.