Question

$$\text {Evaluate each series or state that it diverges.}$$ $$\sum_{k=1}^{\infty} \frac{\pi^{k}}{e^{k+1}}$$

Short answer

Answer: The series converges.

Step-by-step solution

Write down the ratio test formula

We will use the ratio test to determine whether the given series converges or diverges. The ratio test is given by the formula: $$\lim_{k \to \infty} \frac{a_{k+1}}{a_k},$$ where \(a_k\) represents the \(k\)th term of the series. If the limit is less than 1, the series converges. If it is equal to or greater than 1, the series diverges.

Calculate the ratio of consecutive terms

First, we find the \((k+1)\)th term, \(a_{k+1}\), by replacing \(k\) with \(k+1\): $$a_{k+1} = \frac{\pi^{k+1}}{e^{(k+1)+1}} = \frac{\pi^{k+1}}{e^{k+2}}.$$ Now, calculate the ratio \(\frac{a_{k+1}}{a_k}\): $$\frac{a_{k+1}}{a_k} = \frac{\frac{\pi^{k+1}}{e^{k+2}}}{\frac{\pi^k}{e^{k+1}}} .$$

Simplify the expression

To simplify the expression, we need to multiply the numerator and denominator by \(e^{k+1} \pi^k\): $$\frac{a_{k+1}}{a_k} = \frac{\pi^{k+1} e^{k+1}}{e^{k+2} \pi^k} .$$ Now, we can simplify the expression by canceling out the common terms: $$\frac{a_{k+1}}{a_k} = \frac{\pi}{e} .$$

Test for convergence

We found that the limit of the ratios of consecutive terms is constant and equal to \(\frac{\pi}{e}\). Since this value is greater than 0 and less than 1 (<1): $$\frac{\pi}{e} < 1 .$$ By the ratio test, the given series converges.

Key concepts

Ratio Test

The ratio test is a powerful tool for determining the convergence or divergence of an infinite series. It simplifies the process by focusing on the behavior of the series' terms as the number of terms, denoted by \(n\), approaches infinity. To apply the ratio test, we follow these steps:

• Identify the general term \(a_n\) of the series.
• Find the ratio of the next term to the current term, \(\frac{a_{n+1}}{a_n}\).
• Take the limit of this ratio as \(n\) tends to infinity: \[\lim_{n \to \infty} \frac{a_{n+1}}{a_n}.\]
• If the limit is less than 1, the series is convergent. If it equals 1, the test is inconclusive. If it is greater than 1, the series diverges.

For example, in our exercise, we calculated this limit and found it to be \(\frac{\pi}{e}\). Because \(\pi/e < 1\), the series converges according to the ratio test.
The simplicity of using the ratio test lies in its focus on the relation between consecutive terms, making it efficient for identifying tendencies like convergence.

Infinite Series

An infinite series is the sum of an infinite sequence of numbers, typically expressed as \(\sum_{k=1}^{\infty} a_k\). These series can either converge to a certain value or diverge, meaning they grow without bound.

• When viewed finite terms are summed, they represent a partial sum.
• The challenge is to determine whether adding more terms gets the sum closer to a finite value or not.

Infinite series are central to calculus and analysis, featuring in problems ranging from approximations to physics phenomena. Understanding series behavior through convergence tests, like the ratio test, is crucial for evaluating their meaningful contributions.
In the context of our given series, \(\sum_{k=1}^{\infty} \frac{\pi^k}{e^{k+1}}\), it was essential to determine whether the progressive sum of these terms would approach a specific finite value or continue to rise indefinitely.

Mathematical Convergence

Mathematical convergence refers to the idea that as more terms of a sequence or series are considered, they approach a specific finite limit. Understanding convergence is crucial to ensure series aren't misused in calculations.

• If a sequence or series converges, its terms get closer and closer to a certain number.
• Denoting the limit as \(L\), as \(n\) increases, \(a_n\) should satisfy the condition \(\lim_{n \to \infty} a_n = L\).

In our context, convergence tells us whether it is possible to replace an infinite series with a manageable value that approximates the whole series. Divergent series do not have a finite sum because their terms eventually stop getting closer to any particular number.
In practice, understanding whether a series converges allows us to confidently use sums of infinite terms in real-world applications. In our exercise, verifying that \(\frac{\pi^k}{e^{k+1}}\) led to convergence using the ratio test was key in determining its limit behavior and making rigorous predictions in calculations.