Problem 22 - Calculus: One and Several Variables

Question

Show that the series diverges. $$\sum_{k=2}^{\infty} \frac{k^{k-2}}{3 k}$$

Step-by-Step Solution

Answer

For the given series \(\sum_{k=2}^{\infty} \frac{k^{k-2}}{3k}\), we first define the general term \(a_k = \frac{k^{k-2}}{3k}\). We then calculate the ratio of consecutive terms, \(\frac{a_{k+1}}{a_k}\) and simplify it to \(\frac{(k+1)^{(k-1)}}{k^{k-2}(k+1)}\). Applying the Ratio Test, we determine that if the limit of this ratio is greater than 1 as \(k\) approaches infinity, the series diverges. Taking the limit of this ratio and rearranging it, we end up with \(e\) (which is greater than 1), logically indicating that series diverges.

Step 1: Define the general term of the series

We are given the series: \(\sum_{k=2}^{\infty} \frac{k^{k-2}}{3k}\) Let the general term of the series be given by: \(a_k = \frac{k^{k-2}}{3k}\)

Step 2: Calculate the ratio of consecutive terms

Now calculate the ratio of consecutive terms, \(\frac{a_{k+1}}{a_k}\) for the summation: \(\frac{a_{k+1}}{a_k} = \frac{\frac{(k+1)^{(k+1)-2}}{3(k+1)}}{\frac{k^{k-2}}{3k}}\)

Step 3: Simplify the ratio

By simplifying the expression, we get: \(\frac{a_{k+1}}{a_k} = \frac{(k+1)^{(k-1)}}{k^{k-2}(k+1)}\)

Step 4: Apply the Ratio Test

Based on the Ratio Test, if the limit of the ratio of consecutive terms is greater than 1 as \(k\) approaches infinity, then the series diverges: \(\lim_{k \to \infty} \frac{a_{k+1}}{a_k} = \lim_{k \to \infty} \frac{(k+1)^{(k-1)}}{k^{k-2}(k+1)}\)

Step 5: Manipulate the limit expression

Take the limit of the expression as \(k\) approaches infinity: \[\lim_{k \to \infty} \frac{(k+1)^{(k-1)}}{k^{k-2}(k+1)}\] Rearrange expression to have a more meaningful limit: \[\lim_{k \to \infty} \frac{(k+1)^{(k-1)}}{k^{k-1}} = \lim_{k \to \infty} \left(\frac{k+1}{k}\right)^{k-1}\] By taking the limit as \(k\) approaches infinity, it approaches \(e (e > 1)\): \[\lim_{k \to \infty} \frac{(k+1)^{(k-1)}}{k^{k-1}} = e\]

Step 6: Conclude

Since the limit of the ratio of consecutive terms is equal to \(e\) (which is greater than 1), we can conclude that the series \(\sum_{k=2}^{\infty} \frac{k^{k-2}}{3k}\) diverges.

Key Concepts

Ratio Test

The Ratio Test is a powerful tool in calculus to determine the convergence or divergence of infinite series. It involves taking the limit of the ratio of consecutive terms in the series. Specifically, for a given series \(\sum_{n=1}^{\text{\infty}} a_n\), you calculate \(L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|\). If L < 1, the series converges absolutely. If L > 1, the series diverges. If L = 1, the test is inconclusive, and we must use other methods to determine the series' behavior.

This test is particularly useful because it doesn't require the calculation of the sum of the series to determine its convergence. However, it's important to remember that the Ratio Test can only be applied to series where all terms are non-zero, and the ratio of consecutive terms exists for all sufficiently large n.

Convergent and Divergent Series

In the realm of infinite series, a series can either converge or diverge. A convergent series is one where the sum of its terms approaches a finite limit as you add more and more terms. A classic example is the geometric series \(\sum_{n=0}^{\text{\infty}}ar^n\) for \( |r| < 1\), which converges to \(\frac{a}{1-r}\).

On the other hand, a divergent series is one that does not converge to a finite limit. This could mean that the series grows without bound or that it oscillates without settling down to a single value. Understanding the behavior of a series is crucial in many applications, such as solving differential equations or analyzing signals in engineering.

Limits

Limits are a fundamental concept in calculus and are essential in understanding the behavior of functions as they approach a particular point. They essentially describe what happens to a function's value as its variable approaches a specific number or even infinity. The notation \(\lim_{x \to c} f(x)\) represents the limit of the function f(x) as x gets arbitrarily close to the value c.

Limits allow us to define concepts such as continuity, derivatives, and integrals, which are indispensable in calculus. They also form the basis of many tests for convergence, such as the Ratio Test, where the limit of the ratio of consecutive terms of a series determines its convergence status.

Infinite Series

An infinite series is a summation of a sequence of terms that continues indefinitely. One common example is the sum of the terms of an arithmetic or geometric sequence. Infinite series can have finite sums, as strange as it may sound. This property is what we call convergence. If a series doesn't converge to a finite limit, then it is said to diverge.

Infinite series aren't just theoretical curiosities, they play crucial roles in fields such as physics, engineering, and finance, representing anything from time-series data to power series expressions for functions. Distinguishing between convergent and divergent infinite series is important, as it can inform us about the behavior of functions and the solvability of equations in these fields.