Explicit formula for a convergent series

[trulyfalse]

Hello PF.

Homework Statement

A function f is defined by f(x) = 1 + 2x + x² + 2x³ + x⁴ + 2x⁵ + x⁶... Find the radius of convergence of the series and the explicit formula for f(x).

Homework Equations

The Attempt at a Solution

I know that the formula for the series is going to be similar to the function xⁿ2^c, where c is some expression dependent on n that is equal to 1 when n is odd and equal to 0 when n is 0 or when n is even. However, I'm having difficulty finding what c actually is in terms of n. Maybe there's something obvious that I'm not seeing here?

 

[Dick]

Hello PF.

Homework Statement

A function f is defined by f(x) = 1 + 2x + x² + 2x³ + x⁴ + 2x⁵ + x⁶... Find the radius of convergence of the series and the explicit formula for f(x).

Homework Equations

The Attempt at a Solution

I know that the formula for the series is going to be similar to the function xⁿ2^c, where c is some expression dependent on n that is equal to 1 when n is odd and equal to 0 when n is 0 or when n is even. However, I'm having difficulty finding what c actually is in terms of n. Maybe there's something obvious that I'm not seeing here?

Write it as (1+2x)+(1+2x)x^2+(1+2x)x^4+...Now do you see what you are missing?