Sum of infinite geometric series with two terms in summation

Question

I have an infinite geometric series: $$\sum_{i=1}^{\infty} \frac{r^i}{(1+d)^i},$$ where $d$ is a constant. I would like to use the sum of infinite geometric series formula, but I cannot see how to use it because of the $(1+d)^i$ term. How could I calculate this sum?

Thanks.

Answer 1

Note that the summation can be written as

$$\sum_{i = 1}^{\infty}\bigg(\frac{r}{1+d}\bigg)^i = \sum_{i = 1}^{\infty}\bigg(\frac{r}{1+d}\bigg)\cdot\bigg(\frac{r}{1+d}\bigg)^{i-1}$$

and that

$$\sum_{i = 1}^{\infty}u_1\cdot r^{i-1} = \frac{u_1}{1-r}$$

given that $\vert r\vert < 1$.

Answer 2

Express it as $$ \sum_{i=1}^\infty \left( \frac{r}{1+d}\right)^i $$ which is geometric, albeit with a different common ratio.