Finding the Radius and Interval of Convergence for the Summation Series: ∑(n=1 to ∞) [(x+2)^n / (n*4^n)]

As an excellent high school mathematics teacher, I will guide you through the process of finding the radius and interval of convergence for the given summation series.

Step 1: Identify the general term of the series

The general term of the given series is:

a_n = (x+2)^n / (n*4^n)

Step 2: Determine the radius of convergence

To find the radius of convergence, we can use the ratio test. The ratio test states that if the limit of the ratio of consecutive terms, |an+1 / an|, as n approaches infinity, is less than 1, then the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive.

Let's calculate the ratio of consecutive terms:

|an+1 / an| = |(x+2)^(n+1) / ((n+1)4^(n+1)) / ((x+2)^n / (n4^n))| = |(x+2) / (n+1) * 4^(-1)|

Taking the limit as n approaches infinity, we get:

lim (n->∞) |an+1 / an| = lim (n->∞) |(x+2) / (n+1) * 4^(-1)| = |(x+2) / 4^(-1)| = |4 / (x+2)|

Now, we need to determine the value of x for which the limit is less than 1. This will give us the radius of convergence.

|4 / (x+2)| < 1 4 / (x+2) < 1 x+2 > 4 x > 2

Therefore, the radius of convergence is 2.

Step 3: Determine the interval of convergence

The interval of convergence is the set of all values of x for which the series converges. Based on the radius of convergence, the interval of convergence is:

(-2, ∞)

This means that the series converges for all values of x greater than -2.

In conclusion, the radius of convergence for the given summation series is 2, and the interval of convergence is (-2, ∞).