What is the ratio test for proving absolute convergence of a series?

[steven187]

hello all

well i think I am kind of brain dead, iv been workin on a lot of problems over the last few days, I can't 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]

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]

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.