For the following power series, state the intervals of convergence and radius of convergence

Question

Can you detail how to set up the radiance of convergence and the radius:

∞

1) ∑ k!x^k

k=0

∞

2) ∑ (x-3)^k/k

k=0

∞

3) ∑ (-3)^kx^k/√k+1

k=0

Answer 1

Usually (and certainly for the problems you have submitted) you determine the radius of convergence from the ratio. test for convergence.

The absolute value of ratio of the (k+1)^th term to the k^th term must be less than 1 for convergence.

The first series doesn't converge.

Be careful with the second series because the absolute value is important. It has an interval of convergence only.

The answer in the 3^rd series is straight forward.