Interval of convergence problem?

Question

Find the interval of convergence for which this series will converge:

A) 1 + 2x + 4x^2 + 8x^3 + ...

B) 1 + (2/3)x + (4/9)x^2 + (8/27)x^3 + ...

thanks.

Answer 1

A)

We have:

Σ [n=0 to ∞] (2ⁿ)xⁿ

Hence, we examine:

lim [n→∞] |2ⁿ/2ⁿ⁺¹| =

lim [n→∞] ½ =

½ .

Hence, the radius of convergence is ½, and by quick inspection of the endpoints, we see that the interval of convergence is (-½,½).

B)

We have:

Σ [n=0 to ∞] ((⅔)ⁿ)xⁿ

Hence, we examine:

lim [n→∞] |(⅔)ⁿ/(⅔)ⁿ⁺¹| =

lim [n→∞] 1/(⅔) =

3/2 .

Hence, the radius of convergence is 3/2, and by quick inspection of the endpoints, we see that the interval of convergence is (-3/2,3/2).