- #1

Potatochip911

- 318

- 3

## Homework Statement

I put up the image so that you can see the hints if you're curious. I am supposed to prove that if ## S=\sum_{n=0}^{\infty}a_{n}x^{n}## converges for ##|x|<1##, and if ##|a_{n+1}|<|a_{n}|## for ##n>N##, then $$|S-\sum_{n=0}^{N}a_{n}x^{n}|<|a_{N+1}x^{N+1}|\div (1-|x|)$$

## Homework Equations

3. The Attempt at a Solution [/B]

$$S=\sum_{n=0}^{N}a_{n}x^{n}+\sum_{N+1}^{\infty}a_{n}x^{n} \\

S\leq \sum_{n=0}^{N}a_{n}x^{n}+|a_{N+1}x^{N+1}|+|a_{N+2}x^{N+2}|+\cdots \\

S<\sum_{n=0}^{N}a_{n}x^{n}+|a_{N+1}x^{N+1}|+|a_{N+1}x^{N+2}+\cdots \\

S<\sum_{n=0}^{N}a_{n}x^{n}+(|a_{N+1}|)(|x^{N+1}|+|x^{N+2}|+\cdots) \\

S-\sum_{n=0}^{N}a_{n}x^{n}<(|a_{N+1}|)(\frac{1}{1-|x|})

$$

So as you can see I haven't quite gotten it, the main problem is that I'm missing the ##x^{N+1}## on the RHS.