Problem 14 - Calculus: One and Several Variables

Question

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

Step-by-Step Solution

Answer

The interval of convergence for the series \(\sum \frac{1}{\ln k} x^{k}\) is \((-1, 1)\).

Step 1: Recall the Ratio Test

The ratio test states that for a series \(\sum a_n x^n\), the ratio \(|a_{n+1} / a_n| < 1\) indicates that the series is convergent. Conversely, if \(|a_{n+1} / a_n| > 1\) the series is divergent.

Step 2: Apply the Ratio Test

For the given series \(\sum \frac{1}{\ln k} x^{k}\), we want to find \(\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\). This equates to \(\lim_{k \to \infty} \left| \frac{\frac{1}{\ln (k+1)} x^{k+1}}{\frac{1}{\ln k} x^k} \right| = \lim_{k \to \infty} \left| \frac{\ln k}{\ln (k+1)} x \right|\).

Step 3: Simplify the Limit

As \(k \to \infty\), the ratio \(\frac{\ln k}{\ln (k+1)} \to 1\). Therefore, our ratio test simplifies to \(|x| < 1\).

Step 4: Check Boundaries

We need to check our boundaries \(x=1\) and \(x=-1\) separately. Due to ln(k) increasing very slowly for large values of \(k\), the original series doesn't converge at \(x=1\) and \(x=-1\).

Key Concepts

Ratio Test

When it comes to understanding the convergence of series in calculus, the ratio test is a useful tool to determine if a series changes direction—or, in more formal terms, whether it converges or diverges. The ratio test involves studying the ratio of successive terms in a series.
To apply the test to a series \(\sum a_n x^n\), we measure the limit \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\). If this limit is less than 1, the series converges. On the other hand, if the limit is greater than 1, the series diverges.
For the exercise \(\sum \frac{1}{\ln k} x^{k}\), this means calculating \(\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\) where \(a_k = \frac{1}{\ln k}\). Simplifying each part helps us see if the series behaves nicely (i.e., converges) as \(k\) grows large. This method is both simple and powerful, especially when traditional convergence techniques become cumbersome.

Convergence of Series

Understanding why a series converges is essential for fully grasping calculus. Convergence means that as you keep adding more and more terms, the series settles down to a single number instead of drifting off to infinity.
When using the ratio test, analyzing the limit of the ratio of successive terms can provide insight. For a series like \(\sum \frac{1}{\ln k} x^{k}\), the test leads us to consider the expression \(|x| < 1\) as a condition for convergence.

• If \(|x|\) is less than 1, the terms in the series become smaller until they barely change the sum, indicating convergence.
• Conversely, if \(|x|\) is greater than or equal to 1, the terms do not shrink appropriately, leading to divergence.

This understanding is crucial because it underlies many applications in mathematics and science, where knowing whether a total "sum" makes sense or not is foundational.

Interval of Convergence

The interval of convergence pinpoints exactly which values of \(x\) make a series converge. For power series, it is especially significant since such series only behave predictably within certain ranges of \(x\).
After applying the ratio test, as seen with \( \sum \frac{1}{\ln k} x^{k}\), we determine that \(|x| < 1\). This means the series converges for all \(x\) values between -1 and 1 but requires examining boundary points separately.

• At \(x = 1\), as well as \(x = -1\), specific tests must be applied to see if these points fit within our interval.
• For the series in the problem, checking these boundaries reveals that the series doesn't converge at these points due to the slowly increasing nature of \(\ln(k)\).

Thus, the actual interval of convergence doesn't include the endpoints \(-1\) and \(1\), constrained to \((-1, 1)\). Understanding these boundaries is critical for identifying when a series is predictable and usable, especially in real-world applications like physics and engineering.