91Ó°ÊÓ

A power series is essentially a series where each term involves a variable raised to increasing powers. It's written in the general form:

• \( a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots \)

Here, each coefficient \( a_n \) can vary, and the variable \( x \) can take various values. Thus, a power series can be understood as a type of infinite polynomial. In the given exercise, the power series is \( \sum_{n=1}^{\infty} \frac{(\ln n) x^{n}}{3^{n}} \), characterized by the variable \( x \) and coefficients \( \frac{(\ln n)}{3^n} \).
Power series are valid for a range of values known as the interval of convergence. Understanding this range is key to knowing where the series behaves properly and converges to a sum.

The Ratio Test is a useful tool for determining the convergence of a series, especially power series. The essence of the test is to analyze the limit of the ratio between successive terms as \( n \rightarrow \infty \). In equation form:

• \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \)

For the given power series, the test was applied by calculating the limit:

• \( \left( \frac{\ln(n+1)}{\ln n} \right) \left( \frac{x}{3} \right) \)

As \( n \to \infty \), it simplifies to \( \left| \frac{x}{3} \right| \). The series will converge if this result is less than 1: \( \left| \frac{x}{3} \right| < 1 \). This conclusion helps in identifying the basic range within which \( x \) will allow the series to sum to a finite number.

The convergence interval of a power series is a range of the variable \( x \) values where the series converges. In simple terms, it tells us where the series makes mathematical sense. For the given exercise, after applying the Ratio Test, we concluded that:

• \(-3 < x < 3\)

This range states the open interval within which the power series converges. However, evaluating convergence at the endpoints \( x = -3 \) and \( x = 3 \) is crucial, since these may or may not cause the series to diverge. When these endpoints were checked, both led to series that diverged, so they are not included in the final interval of convergence. Thus, the interval of convergence is strictly between \(-3\) and \(3\) without including the endpoints.

Divergence in the context of a series occurs when its terms do not sum to a finite value. This can happen when the series doesn't conform to convergence criteria across all its possible values. In the case of our power series exercise:

• At \( x = 3 \), the test evaluated the series \( \sum_{n=1}^{\infty} \frac{(\ln n)}{n} \) and found that it diverged using the integral test.
• At \( x = -3 \), the series \( \sum_{n=1}^{\infty} \frac{(-1)^n (\ln n)}{n} \) also diverged, this time by the limit comparison test with a similar divergence characteristic to the harmonic series.

These findings confirm that neither endpoint should be included within the interval of convergence. Divergence checks are pivotal in determining the precise boundaries of convergence intervals for power series.