# absolute convergence

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• Nov 26th 2007, 11:18 AM
taypez
absolute convergence
Suppose that a_k >= 0 and a_k^(1/k) → a as k → ∞. Prove that ∑ (k=1 to ∞) a_kx^k converges absolutely for all abs (x) < 1/a if a ≠ 0 and for all x in R if a=0.

Thanks.
• Nov 26th 2007, 02:51 PM
Plato
Quote:

Originally Posted by taypez
Suppose that a_k >= 0 and a_k^(1/k) → a as k → ∞. Prove that ∑ (k=1 to ∞) a_kx^k converges absolutely for all abs (x) < 1/a if a ≠ 0 and for all x in R if a=0.

This is an odd question I think.
It is nothing more than a restatement of the root test.
• Nov 26th 2007, 06:04 PM
ThePerfectHacker
Quote:

Originally Posted by taypez
Suppose that a_k >= 0 and a_k^(1/k) → a as k → ∞. Prove that ∑ (k=1 to ∞) a_kx^k converges absolutely for all abs (x) < 1/a if a ≠ 0 and for all x in R if a=0.

Thanks.

If $\displaystyle |x| < \frac{1}{a}$ (assuming $\displaystyle a\not =0$ so $\displaystyle a>0$) then $\displaystyle \limsup |a_kx^k|^{1/k} = \limsup |a_k| \cdot |x| = \frac{1}{a}|x| < 1$ so it converges absolutely.