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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