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.
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.
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.