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". 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

User Toshkuuu
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1 Answer

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Final answer:

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.

User Bikas Lin
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