The power series has a radius of convergence R = 2. Determine the interval of convergence.

Question

The power series  has a radius of convergence R = 2. Determine the interval of convergence.

	(-1, 3)
	[-1, 3]
	(-1, 3]
	[-1, 3)

Answer 1

At the endpoints, x = 3 and x = -1, the series basically becomes 3 times the harmonic series, since the numerator is either 3·2ⁿ or 3·(-2)ⁿ, which cancels with the 2ⁿ in the denominator. However, since the harmonic series is only conditionally convegent (ie when the signs of the terms alternate), the series converges for x = -1 but diverges for x = 3. Thus, the interval of convergence is [ -1 , 3 ).