Thread: 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.

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.

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 (assuming so ) then so it converges absolutely.