Can someone explain why this is a proof?

[flyingpig]

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?

 

[Dick]

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]

There is an error in what you wrote.

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

You should have
$$As\;n\to\infty\;,r^{n+1}\to 0$$

 

[flyingpig]

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