Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(\sum_{k=1}^{\infty} \frac{k^{2}}{e^{k}}\)

Step-by-Step Solution

Answer

The series \(\sum_{k=1}^{\infty} \frac{k^{2}}{e^{k}}\) converges.

Step 1 - Verify Convergence of the Series

To determine if the series \(\sum_{k=1}^{\infty} \frac{k^{2}}{e^{k}}\) converges, we must first verify if it converges. Because \(e^{k}\) grows exponentially, it outgrows any polynomial function such as \(k^2\). We apply the ratio test to confirm.

Step 2 - Apply the Ratio Test

The ratio test states that for a series \(\sum_{k=1}^{\infty} a_k\), if the limit \(\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| < 1\), then the series converges. Here, \(a_k = \frac{k^2}{e^k}\).Calculate the ratio:\[\left| \frac{a_{k+1}}{a_k} \right| = \left| \frac{(k+1)^2 e^k}{k^2 e^{k+1}} \right| = \frac{(k+1)^2}{k^2 e}\]Factor and simplify:\[\lim_{k \to \infty} \frac{k^2 + 2k + 1}{k^2 e} = \frac{1}{e}\]Since \(\frac{1}{e} < 1\), the series converges by the ratio test.

Step 3 - Identify the Type of Convergent Series

The series is a power series with exponential terms. Its convergence is guaranteed, but it does not match a simple geometric or arithmetic format that provides a straightforward sum. Instead, it resembles a type where special functions (like the polylogarithm) or computational tools are used for an exact evaluation.

Key Concepts

Ratio Test

The ratio test is a powerful method for determining the convergence of infinite series. To apply it, consider a series \( \sum_{k=1}^{\infty} a_k \). Calculate the limit of the absolute value of the ratio between consecutive terms: \(\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|. \)

• If this limit is less than 1, the series converges.
• If it's greater than 1, the series diverges.
• When it equals 1, the test is inconclusive.

In our exercise, we have \(a_k = \frac{k^{2}}{e^k} \). Substituting into the ratio formula results in a simplification to \( \frac{1}{e} \), which is less than 1. This confirms the series converges.
Utilizing the ratio test is often straightforward and involves evaluating a single limit, making it a preferred choice for quickly assessing convergence of series with terms involving polynomials and exponentials.

Exponential Growth

Exponential growth describes a process where a quantity increases at a constant multiplicative rate. In mathematics, this is depicted as \( e^x \), where \( e \) is Euler's number, approximately 2.71828.
Compared to exponential functions, polynomial functions, like \( k^2 \), grow much more slowly. That's because exponential growth involves repeated multiplication, expanding rapidly as \( x \) increases.

• For example, \( e^k \) grows much faster than \( k^2 \).
• This rapid growth rate is why exponential components dominate over polynomial ones in series expansions, as seen in our given series.

In the exercise, \( e^k \) is in the denominator, ensuring that each term of the series diminishes significantly as \( k \) increases, which aids in proving its convergence.

Series Convergence

Series convergence refers to whether the sum of an infinite series approaches a finite limit. In other words, as you sum more terms in the series, does it lead to a finite result?

• A series that converges will have its partial sums tend towards a specific value as more terms are added.
• A divergent series does not settle towards any particular limit.

In terms of mathematical tests, like the ratio test discussed earlier, they help determine if a series converges without needing to compute the entire sum.
Our specific series \( \sum_{k=1}^{\infty} \frac{k^{2}}{e^{k}} \) converges due to the rapid growth rate of the denominator, surpassing the growth of the numerator. Each term becomes increasingly small, leading the entire series to converge to a finite value.

Power Series

A power series is an infinite series of the form \( \sum_{k=0}^{\infty} a_k x^k \, \) where \( a_k \) are coefficients and \( x \) is a variable. Power series are foundational in calculus for functions' approximations.

• They generalize polynomials, allowing for potentially complex functions of \( x \).
• Understanding where a power series converges is essential, as it defines the interval of convergence in which the series represents a function accurately.

While the specific series in our exercise doesn't align perfectly with a simple power series, it can be examined using similar ideas.
Recognizing a series as a type of power series helps us explore more sophisticated mathematical tools, like special functions or computational algorithms, to determine sums or behavior of the function represented by the series.