87. Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.


e. ∑ (k = 1 to ∞) (π / e)⁻ᵏ is a convergent geometric series.

Verified step by step guidance

1. Identify the general term of the series: the series is given by \( \sum_{k=1}^{\infty} \left( \frac{\pi}{e} \right)^{-k} \). This can be rewritten as \( \sum_{k=1}^{\infty} \left( \frac{e}{\pi} \right)^k \) because raising to the power \(-k\) is the same as taking the reciprocal and raising to the positive power \(k\).
2. Recognize that this is a geometric series with the first term \( a = \left( \frac{e}{\pi} \right)^1 = \frac{e}{\pi} \) and common ratio \( r = \frac{e}{\pi} \).
3. Recall the convergence criterion for a geometric series: a geometric series \( \sum a r^{k-1} \) converges if and only if \( |r| < 1 \).
4. Evaluate the absolute value of the common ratio: since \( e \approx 2.718 \) and \( \pi \approx 3.1415 \), \( \left| \frac{e}{\pi} \right| < 1 \).
5. Conclude that because \( |r| < 1 \), the series \( \sum_{k=1}^{\infty} \left( \frac{e}{\pi} \right)^k \) is a convergent geometric series.

Key Concepts

Geometric Series

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio r. It has the form ∑ ar^k, where a is the first term and r is the common ratio. Understanding the structure helps identify if a series fits this pattern.

Convergence of Geometric Series

A geometric series converges if and only if the absolute value of the common ratio |r| is less than 1. When this condition holds, the infinite sum approaches a finite limit given by a/(1-r). If |r| ≥ 1, the series diverges.

Evaluating the Common Ratio

To determine convergence, it is essential to correctly identify and evaluate the common ratio r in the series. In this problem, the term (π/e)^(-k) can be rewritten as (e/π)^k, so the ratio is e/π. Comparing |e/π| to 1 determines if the series converges.