Problem 15 - Calculus: One and Several Variables

Question

Find the interval of convergence. $$\sum \frac{k-1}{k} x^{k}$$

Step-by-Step Solution

Answer

Based on Ratio Test alone, the convergence interval of the series couldn't be determined as the value of \( L = 1 \), and the test is inconclusive. Other convergence/divergence tests, or more detailed inspection of the series' endpoints, could potentially yield further insights about the interval of convergence.

Step 1: Identify the power series

The given series is \(\sum_{k=1}^{\infty} \frac{k-1}{k} x^{k}\) . It's a power series in the form of \( \sum a_k x^k \) where \( a_k = \frac{k-1}{k} . \)

Step 2: Apply the Ratio Test formula

According to the Ratio Test, we find the limit of the ratio of the \((k+1)^{th}\) term to the \(k^{th}\) term for the \(a_k\) portion of the series. Here, this would look like: \( L = \lim_{k \rightarrow \infty} \frac{a_{k+1}}{a_k} \). For the given series, this simplifies to \( L = \lim_{k \rightarrow \infty} \frac{(k+1-1)/(k+1)}{(k-1)/k} = \lim_{k \rightarrow \infty} \frac{k/k+1}{(k-1)/k} \).

Step 3: Simplify the Ratio

Simplify the above ratio and we get \( L = \lim_{k \rightarrow \infty} \frac{k^2}{k^2 -1} = 1.\)

Step 4: Compare L with 1

If L<1, the series converges absolutely. If L>1, the series diverges. If L=1, the test is inconclusive and we need to use other methods. In this case L=1, so we can make no definitive conclusion about the series based on this test alone.

Step 5: Use other methods

Although we can use other convergence/divergence tests when the ratio test is inconclusive, generally power series convergence needs to be established by identifying the radius of convergence first and then checking the endpoints. Here, however, because we have an inconclusive ratio test, it's hard to determine the interval of convergence.

Key Concepts

Power Series

A power series is simply a sum of terms in the form of \(a_k x^k\), where each term is dependent on a variable, usually \(x\). In this context, \(a_k\) represents the coefficients, and \(x^k\) is the variable raised to the power of \(k\), starting from \(k = 0\) and extending to infinity. In the exercise, we are dealing with a specific power series:

• The given series is \(\sum \frac{k-1}{k} x^k\).
• Here, each term of the series is specifically dependent on both the variable \((x)\) and the coefficient \((a_k = \frac{k-1}{k})\).

This form of series is useful because it can be manipulated through different tests to establish convergence properties across varying intervals. By understanding the behavior of \(a_k\) and how it impacts the series, we can determine if and where the series converges.

Ratio Test

The Ratio Test is a fundamental method for determining the convergence of an infinite series. When given an infinite series, the Ratio Test checks the limit of the absolute value of the ratio of consecutive terms:

• We take the limit \(L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\).
• If \(L < 1\), the series converges absolutely.
• If \(L > 1\), the series diverges.
• If \(L = 1\), the test is inconclusive.

In the exercise, using the Ratio Test yields an \(L = 1\), which is inconclusive. This means we cannot make definitive statements about convergence or divergence solely based on this test. This prompts the need for additional methods to evaluate the convergence of the power series.

Radius of Convergence

The radius of convergence is vital in understanding the domain where a power series converges. It is expressed as a non-negative number that defines a range around the center of a power series:

• A power series converges within the interval \(|x| < R\), where \(R\) is the radius of convergence.
• If \(R = 0\), it converges only at \(x = 0\).
• If \(R = \infty\), it converges for all \(x\).

After applying a test like the Ratio Test, even when inconclusive, understanding the radius of convergence gives insight into where the series converges. This is crucial as, in our exercise, it tells us areas to focus on when other tests yield inconclusive results or when assessing endpoints within an interval.

Convergence and Divergence Tests

When the Ratio Test alone does not suffice to determine convergence or divergence (as it was inconclusive in our exercise), other tests must be considered. Some of these tests include:

• **Root Test:** Similar to the Ratio Test but uses roots instead of ratios. Useful when dealing with powers of \(x\).
• **Integral Test:** Applies to series where terms from a function can be integrated over a domain to check for convergence.
• **Alternating Series Test:** Applies when series terms alternate in sign, and additional conditions about term size are met.
• **Comparison Test:** Involves comparing a troublesome series to a known benchmark series.

These tests, among others, allow us to delve deeper and analyze series that do not straightforwardly yield to initial tests like the Ratio Test. Having a variety of methods at our disposal ensures a comprehensive examination of each specific series scenario.