Question

Radius of convergence of series r sum n=0 to infinity (6^n n^6)x^n

Answer 1

To determine the radius of convergence for the power series, the ratio test is applied, simplifying the expression and finding the limit to ascertain the constraint on x for which the series converges.

Step-by-step explanation:

The student is asking about the radius of convergence for the power series ∑ n=0 to infinity (6^n n^6)x^n. In order to find the radius of convergence, we can use the ratio test, which involves looking at the limit as n approaches infinity of the absolute value of a_(n+1)/a_n, where a_n represents the nth term of the series. In the given series, a_n = 6^n n^6 x^n. Calculating this limit will show how x must be constrained for the series to converge.

Applying the ratio test, we get:

1. Find the absolute value of the ratio of consecutive terms, which is |(6^(n+1) (n+1)^6 x^(n+1))/(6^n n^6 x^n)|.
2. Simplify the expression to |6 (n+1)^6 x/n^6|.
3. Take the limit as n approaches infinity of the simplified expression.
4. The series converges when this limit is less than 1, which provides us with an inequality to find the radius of convergence for x.