Question

Find the radius of convergence and interval of convergence for the power series ( ∑n=0^[infinity] (3ⁿ ⋅ xⁿ ⋅ (n+1)) ).

a) ( |x| < 1/3 )
b) ( |x| < 1/2 )
c) ( |x| < 1 )
d) ( |x| < 2 )

Answer 1

To find the radius of convergence and interval of convergence for the given power series, use the ratio test. The radius of convergence is 1/3 and the interval of convergence is (-1/3, 1/3).

Step-by-step explanation:

To find the radius of convergence and interval of convergence for the given power series, we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms approaches a positive number less than 1, then the power series converges. If the limit is greater than 1, then the power series diverges. In this case, we have the power series ( ∑n=0^[infinity] (3ⁿ ⋅ xⁿ ⋅ (n+1)) ).

Using the ratio test, we calculate:

|(3^(n+1) ⋅ x^(n+1) ⋅ (n+2)) / (3ⁿ ⋅ xⁿ ⋅ (n+1))| as n approaches infinity.

Simplifying the expression, we get |3x(n+1) / 3ⁿxⁿ| , which simplifies further to |x(n+1) / 3ⁿxⁿ| .

Taking the limit as n approaches infinity, we get |x / 3x| = |1 / 3|.

Since the limit is less than 1, the power series converges. Therefore, the radius of convergence is 1/3 and the interval of convergence is (-1/3, 1/3).