# Can someone explain why this is a proof?

## Homework Statement

Prove that the geometric series $$\sum_{n=1}^{\infty} r^n$$ if -1 < r < 1

2. The Solution

$$s_n = r + r^2 + ... + r^n$$

$$rs_n =r^2 + r^3 ... + r^{n+1}$$

$$s_n - rs_n = r - r^{n+1}$$

$$s_n = \frac{r - r^{n+1}}{1 -r}$$

For |r|<1

$$As\;n\to\infty\;,r^{n+1}\to \infty$$

Therefore

$$\lim_{n\to\infty}s_n=\frac{r}{1-r}$$

Q.E.D

Question

The solution is what we took in notes during lecture.

Now here is my question why does $$\lim_{n\to\infty}s_n=\frac{r}{1-r}$$ answer the proof? How does that prove the geometric series $$\sum_{n=1}^{\infty} r^n$$ converge?

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Dick
Homework Helper
Because s_n is a partial sum of the series. And the definition of convergence of a series is that the limit of the partial sums converges. Look up the definition of convergence.

Mark44
Mentor
There is an error in what you wrote.
|r| < 1
$$As\;n\to\infty\;,r^{n+1}\to \infty$$
You should have
$$As\;n\to\infty\;,r^{n+1}\to 0$$

There is an error in what you wrote.

You should have
$$As\;n\to\infty\;,r^{n+1}\to 0$$
Yeah lol, it seems I make one of these TEX mistakes every time lol

thanks for catching that