# Absolutely convergent

#### steven187

hello all

well i think im kind of brain dead, iv been workin on alot of problems over the last few days, I cant see anything obvious anymore, well this shall be the last one for today (i hope), anyway here it is,

suppose that for some $$x\not= 0$$, the series
$$\sum_{n=1}^{\infty} a_n x^n$$
is convergent. Prove the series is absolutely convergent for all $$w$$ with $$|w|<|x|$$.

Steven

#### Muzza

Well, a power series diverges outside its radius of convergence and converges absolutely on the inside...

#### Icebreaker

Try the squeeze theorem. What's with a_n, though? You mean each term has a different coefficient?

#### HallsofIvy

Homework Helper
Icebreaker said:
Try the squeeze theorem. What's with a_n, though? You mean each term has a different coefficient?
Well, yes! That is the basic idea of a power series after all.

#### Icebreaker

Odd, I have the idea in my head that they must have the same coefficient in order to find its sum, if it's convergent.

#### saltydog

Homework Helper
Hello Steven.

How about using the ratio test:

If:

$$\mathop\lim\limits_{n\to\infty} |\frac{u_{n+1}}{u_n}|=L<1$$

then the given series is absolutely convergent.

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