91Ó°ÊÓ

Power series are expressions of the form \( \sum_{n=0}^{\infty} a_n (x-c)^n \). These series are centered at a specific point \(c\), and involve terms that become increasingly more complex as \(n\) increases.
In the provided exercise, we are dealing with a power series centered at \(x = 2\), written as:

• \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n \cdot 10^{n}}(x-2)^{n} \)

The coefficients, \(a_n\), in this series are \( \frac{(-1)^{n+1}}{n \cdot 10^{n}} \), and the variable part is expressed as \((x-2)^n\). These elements are crucial in determining the series’ behavior, particularly in terms of convergence and where it converges.

The Ratio Test is a powerful tool used to determine the convergence of a series. It simplifies and quantifies how the terms of a series grow relative to each other.
To apply the Ratio Test, you analyze the series’ terms, \(a_n\), and calculate the \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). This gives a clear criterion:

• If the limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, the series diverges.
• When the limit equals 1, the test is inconclusive.

In our example, we focused on the terms \(a_n = \frac{(-1)^{n+1}}{n \cdot 10^{n}}(x-2)^{n}\) and found:\[ \left| \frac{x-2}{10} \right| < 1. \]This inequality indicates the possible range of \(x\) where the series converges, setting up for further steps.

Understanding convergence means knowing when a series sums up to a finite value. For our power series, this convergence depends on the values of \(x\) in relation to the series' behavior. The derived inequality from the ratio test, \( \left| x-2 \right| < 10 \), translates into the interval \(-8 < x < 12\). This is where the series has the potential to converge.
Add 2 to each term of the inequality to account for the center of the series, ensuring it focuses on its convergence interval related to \(x\).When a series converges, it means the infinite sum of its terms approaches a specific, finite value. This behavior is tested at the interval's endpoints to check whether they are included.

Endpoints testing involves checking the convergence of a series at the borders of its interval, determined through the ratio test.In the solution:

• We tested \(x = -8\) by substituting into the original series, leading to an alternating harmonic series which converges.
• We also tested \(x = 12\), yielding a harmonic series which is known to diverge.

The alternating harmonic series converges because its terms illustrate alternating signs and decreasing magnitude, a characteristic known for converging. On the contrary, the harmonic series diverges due to its terms diminishing too slowly to sum to a finite value.Thus, the series converges at \(x = -8\) but not at \(x = 12\), concluding the interval of convergence to be \([-8, 12)\). This valuable step finalizes the exact range where our power series remains bounded and convergent.