Question

State whether the given \(p\) -series converges. \(\sum_{n=1}^{\infty} \frac{n^{\pi}}{n^{2 e}}\)

Short answer

The series converges because \( 2e - \pi > 1 \).

Step-by-step solution

Identify the General Form of the Series

The given series is \( \sum_{n=1}^{\infty} \frac{n^{\pi}}{n^{2 e}} \). This can be rewritten as \( \sum_{n=1}^{\infty} \frac{1}{n^{2 e - \pi}} \), which is a \( p \)-series in the form of \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). In this context, \( p = 2e - \pi \).

Apply the Convergence Criterion for p-Series

A \( p \)-series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) converges if \( p > 1 \) and diverges if \( p \leq 1 \). Here, we need to determine if \( 2e - \pi > 1 \).

Calculate 2e - π

Compute \( 2e \) and \( \pi \) separately. Using approximations: \( e \approx 2.718 \) and \( \pi \approx 3.141 \). Thus, \( 2e \approx 2 \times 2.718 = 5.436 \). Therefore, \( 2e - \pi \approx 5.436 - 3.141 = 2.295 \).

Decision Based on Calculated Value

Since \( 2e - \pi \approx 2.295 \) and this value is greater than 1, the series \( \sum_{n=1}^{\infty} \frac{1}{n^{2e-\pi}} \) converges according to the \( p \)-series test.

Key concepts

Convergence Criteria

Convergence criteria help us determine if an infinite series will end up at a finite number or just keep growing indefinitely. Infinite series are sums of infinitely many terms. To understand their behavior, mathematicians have developed tests to decide convergence or divergence.
One common criterion for convergence is the **comparison test**. It involves comparing the series in question with another one whose behavior is known. Other tests include the **ratio test** and **root test**, each effective under different conditions.
For our specific case with the series in question, the **p-series test** is the go-to test. It will determine whether the series converges or diverges by evaluating the exponent value, known as **p**.

Series Convergence

Series convergence concerns itself with whether or not an infinite series adds up to a finite value. To dive deeper, let's first picture **series** as a sequence of numbers being added one after another. If the sum eventually approaches a specific number, the series is said to converge. If it keeps increasing or decreasing without bound, it diverges.
Convergence is crucial in mathematics because it indicates stability and predictability in the sum of infinite terms. Imagine trying to balance an equation or calculating probabilities; knowing whether a sum converges assures us we aren't dealing with infinity.
When dealing with the convergence of series like the given **p-series**, we ascertain through specific tests, like the **p-series test**, to validate if the sum behaves nicely and converges.

P-Series Test

The p-series test is a method for determining the convergence of a series that takes the form: \[ \sum_{n=1}^{\infty} \frac{1}{n^p} \] Where **p** is a constant. This test is favorable due to its simplicity when compared to broader convergence tests.
Here's the rule:

• If **p** > 1, the series converges.
• If **p** ≤ 1, the series diverges.

The p-series test is powerful as it quickly provides answers, especially for typical mathematical constants like in our series: \[ \sum_{n=1}^{\infty} \frac{1}{n^{2e-\pi}} \] Here, having calculated that \( 2e - \pi \approx 2.295 \), which is greater than 1, confirms convergence according to the p-series rule.