power series

Question

Determine the interval and radius of convergence for the power series of the summation from k=1 to infinity of (x^k)/(k4^k)

 

Answer 1

The ratio test shows that this series will converge for abs(x/4) <1.    The boundary cases x = 4  and x = -4 need to be checked.   For x = 4, the series is the harmonic series which does not converge.  For x = -4, the series is an alternating sign series which, according to the Leibniz test,  does converge (to - ln(2) in fact).

Thus the interval of convergence is   - 4 ≤ x < 4.   The radius of convergence has to be 4 to match this interval.