# What is the radius of convergence by using the ratio test?

##
1)#sum_(n=0)^(\infty)# (n^3)(x^n)

2) #sum_(n=0)^(\infty)# ((2^n)((x-1)^n))/(n)

1)

2)

#R=1# 2.#R=1/2#

According to the Ratio Test, we should let

Three scenarios can occur when working with Power Series.

Let's now administer the test.

Next, we have

Reduce complexity.

Division is multiplication by the reciprocal, so let's take the limit.

Thus,

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The radius of convergence of a power series can be determined using the ratio test. The ratio test states that if the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term is less than 1, then the series converges absolutely. The radius of convergence (R) is found by taking the limit as n approaches infinity of the absolute value of the ratio of consecutive terms, then applying the properties of limits. Mathematically, R is equal to the limit as n approaches infinity of the absolute value of (a_(n+1) / a_n).

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