Prove that the series 1 2x x2 2x3 anxn where an1 if

Prove that the series 1 + 2x + x^2 + 2x^3 + ? + anx^n + ? where an=1 if n is even (include n=0) and an=2 if n is odd has (-1,1) as its interval of convergence.

Solution

The series can be written as

1+x2+..._x2n+...+

2x(1+x2+..._x2n+...+)

= (1+2x))(1+x2+..._x2n+...+)

=(1+2x) (infinite geometric series with common ratio x^2)

Hence converges if x^2<1 or |x|<1\\

Hence interval of convergence is (-1,1)

and converges to