Explain the following step

Question

In an exercise the following (simplified) step was made, now I can't recall how I came to this, please explain the following step:

$$... = \sum\limits_{m=0}^\infty \frac{1}{1000^m} = \frac{1}{1-\frac{1}{1000}} =... $$

Try to google 'geometric series'. – Matti P. Dec 21 '17 at 09:06 — Matti P., Dec 21 '17 at 09:06
Thank you , I did not view this as a series! – Michelle_B Dec 21 '17 at 09:13 — Michelle_B, Dec 21 '17 at 09:13

score 3 · Answer 1 · answered Dec 21 '17 at 09:16

Recall the known geometric series :

$$\frac{1}{1-w} = \sum_{n=0}^\infty w^{n} \quad \text{for} \quad |w|\leq1$$

Applying this to your given expression for $w = \frac{1}{1000}$ and for $n=m$ :

$$\sum\limits_{m=0}^\infty \frac{1}{1000^m} = \frac{1}{1-\frac{1}{1000}}$$

which holds, since :

$$\bigg|\frac{1}{1000}\bigg| < 1$$

score 1 · Accepted Answer · answered Dec 21 '17 at 09:12

This is an infinite geometric series and the sum to infinity of a geometric series is $\frac{a}{1-r}$, where a is the starting term which here is $\frac{1}{1000^0}=1$, and r is the ratio each term is multiplied by to get the next term which is $\frac{1}{1000}$.

Putting all this together we get $\frac{1}{1-\frac{1}{1000}}$

Hope that helps :)

Explain the following step

2 Answers2