Question

Find the radius of convergence, r, of the series. [infinity] Σ (2nxⁿ)/(n²), n = 1.

a) r = 2
b) r = 4
c) r = 8
d) r = 16

Answer 1

The ratio test is used to find the radius of convergence. The limit is taken and if it is less than 1, the series is convergent.

Step-by-step explanation:

The radius of convergence, r, of the series can be found using the ratio test. According to the ratio test, if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series is convergent. So, let's apply the ratio test to the given series:

Let's take the absolute value of the ratio of consecutive terms:

|(2(n+1)xⁿ⁺¹)/(n⁺¹)²| / |(2nxⁿ)/(n²)|

Simplify the expression:

|2(n+1)/(n+1)²|

Take the limit as n approaches infinity:

lim(n→∞) |2(n+1)/(n+1)²| = lim(n→∞) 2/(n+1) = 0

Since the limit is less than 1, the series is convergent. Therefore, the radius of convergence, r, is d) r = 16.