Question

Find an expression for the power series, then use the ratio test to find the radius of convergence?

`1+2x+(4x^2)/(2!)+ (8x^3)/(3!)+(16x^4)/(4!)+(32x^5)/(5!)+cdots`

Answer 1

Notice the numerators are the powers of `2` starting at `0`. Therefore,

`sum_(n = 0)^oo (2x)^n/(n!)`

The radius of convergence is given by the ratio test:

`L = lim_(n ->oo) ((2x)^(n + 1)/((n + 1)!))/(((2x)^n)/(n!))`

`L = lim_(n -> oo) (2x)/(n + 1)`

`L = 2|x| lim_(n->oo) 1/(n + 1)`

`L = 0`

This means that the sequence converges for all values of `x`.

Hopefully this helps!