Finding the Values of x for Convergence of the Series ∑(5x-9)^n/n^2

As an excellent high school mathematics teacher, I'm excited to share my insights on the topic of finding the values of x for which the infinite series ∑(5x-9)^n/n^2 converges.

Understanding the Series

The given series is an infinite series, where the general term is (5x-9)^n/n^2. To determine the values of x for which this series converges, we need to apply the concepts of series convergence.

Convergence Criteria

The convergence of an infinite series can be determined using various convergence tests, such as the Ratio Test, the Root Test, or the Comparison Test. In this case, we will use the Ratio Test to analyze the convergence of the given series.

The Ratio Test states that if the limit of the ratio of consecutive terms in the series, as n approaches infinity, is less than 1, then the series converges. Mathematically, this can be expressed as:

lim (n→∞) |(a_{n+1} / a_n)| < 1

where a_n represents the nth term of the series.

Applying the Ratio Test

Let's apply the Ratio Test to the given series:

a_n = (5x-9)^n/n^2 a_{n+1} = (5x-9)^(n+1)/(n+1)^2

The ratio of consecutive terms is:

a_{n+1} / a_n = (5x-9)^(n+1)/(n+1)^2 * n^2 / (5x-9)^n = (5x-9) * n^2 / (n+1)^2

Now, we need to find the limit of this ratio as n approaches infinity:

lim (n→∞) |(5x-9) * n^2 / (n+1)^2| = |5x-9|

Convergence Condition

According to the Ratio Test, the series converges if the limit of the ratio is less than 1, which means:

|5x-9| < 1

Solving this inequality, we get:

4 < 5x < 14 4/5 < x < 14/5

Therefore, the values of x for which the series ∑(5x-9)^n/n^2 converges are:

4/5 < x < 14/5

This range of values for x ensures that the infinite series will converge and the sum of the series will be a finite value.