Question

". Consider a power series [infinity] ∑ n=0 anxn. Suppose the limit L =

lim n→[infinity] |an| 1/n exists and is positive. Justify the following
steps, which prove that 1/L is the radius of convergence of the
ser"
A) By the Ratio Test, the radius of convergence is R = 1/L
By the Ratio Test, the radius of convergence is R = 1/L
C) By the Comparison Test, the radius of convergence is R = 1/L
D) By the Integral Test, the radius of convergence is R = 1/L

Answer 1

The radius of convergence R for the power series ∑ n=0 an*x^n, where L = lim n→∞ |an|^(1/n) exists and is positive, is calculated using the Root Test and is given by R = 1/L.

Step-by-step explanation:

The question involves calculating the radius of convergence of a power series using the given limit for the series' coefficients. Specifically, it is stated that the limit L = lim n→∞ |an|^(1/n) exists and is positive. For a power series ∑ n=0 an*x^n, where an represents the coefficients and x is the variable, the radius of convergence R can be determined using the Root Test (also known as the Cauchy-Hadamard formula).

According to the Root Test, if the limit L exists and is positive, then the radius of convergence R is given by R = 1/L. This is because for the power series to converge, the terms must eventually decrease in magnitude, hence the powers of x multiplied by the coefficients must be less than 1 in absolute value. Conveniently, if the limit does not exist or is infinite, the radius of convergence is 0, indicating that the series only converges at x = 0.

So, in this case, since the limit L exists and is positive, the radius of convergence R for the power series indeed equals 1/L. This method is not based on the Ratio, Comparison, or Integral Test, but is sometimes confused with the Ratio Test because both involve limits of ratios of the series' terms.