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