Find the radius of convergence of each of the following power series.

• [a [math]\sum_{i=0}^\infty \frac{x^i}{2^i}[/math]
• [math]\sum_{i=1}^\infty \frac{x^i}{i^2}[/math]
• [math]\sum_{k=1}^\infty \frac{x^k}{\sqrt{k}}[/math]
• [math]\sum_{k=0}^\infty x^k[/math]
• [math]\sum_{k=0}^\infty (-1)^kx^k[/math]
• [math]\sum_{i=0}^\infty 2^iy^i[/math]
• [math]\sum_{i=1}^\infty ix^{i-1}[/math]
• [math]\sum_{n=0}^\infty \frac{y^n}{n(n+1)3^n}[/math].

Find the interval of convergence of each of the power series in Problem Exercise.

Is the following statement true or false: Every power series [math]\sum_{i=0}^\infty a_ix^i[/math] converges absolutely only in the interior of its interval of convergence? Why?

Find the radius of convergence of each of the following power series.

• [math]\sum_{i=1}^\infty \frac{(x-2)^i}{i}[/math]
• [math]\sum_{i=0}^\infty \frac{(x-2)^i}{i!}[/math]
• [math]\sum_{k=0}^\infty \frac{k}{k+1} (x+2)^k[/math]
• [math]\sum_{n=0}^\infty \frac{n!}{2^n} (x-1)^n[/math]
• [math]\sum_{k=1}^\infty \frac{k^2}{5^k} (y+1)^k[/math]
• [math]\sum_{k=0}^\infty (-1)^{k-1} \frac{x^{2k+1}}{(2k+1)!}[/math].

Find the interval of convergence of each of the power series in Problem Exercise.

Prove the following: If [math]\lim_{n\goesto\infty} \frac{|a_{n+1}|}{|a_n|} = \rho[/math], then the radius of convergence of the power series [math]\sum_{i=0}^\infty a_i(x-a)^i[/math] is equal to [math]\frac1{\rho}[/math]. (Assume that [math]\frac10 = \infty[/math] and that [math]\frac1\infty = 0[/math].)