How do I simplify $\frac{10}{(1+ 6\%)} + \frac{10}{(1+ 6\%)^2} +\frac{10}{(1+ 6\%)^3} +... +\frac{10}{(1+ 6\%)^\infty}$

Question

How do I simplify $\frac{10}{(1+ 6\%)} + \frac{10}{(1+ 6\%)^2} + \frac{10}{(1+ 6\%)^3} + ... + \frac{10}{(1+ 6\%)^\infty}$

So I get to $\frac{10}{6\%}$ ?

What are the intermediary steps ? What is the demonstration ?

Answer 1

$$S_n = r + r^2 + ... + r^n$$ Multiply $S_n$ by $r$ and subtract both expresions: $$S_n-S_nr=S_n(1-r)=r-r^{n+1}$$You will obtain $$S_n=\frac{r-r^{n+1}}{1-r}$$

Substitute $r=1/(1+0.06)$ and multiply this result by $10$. In the case that $n\to\infty$ it results that $r^{\infty}\to 0$, and you will simply have: $$S_\infty = \frac{r}{1-r}=\frac{1}{1/r-1}=10\frac{1}{(1+0.06)-1)}$$