Problem 67 - Thomas Calculus

Question

Which series in Exercises \(49-68\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$ \sum_{n=0}^{\infty}\left(\frac{e}{\pi}\right)^{n} $$

Step-by-Step Solution

Answer

The series converges with a sum of \(\frac{\pi}{\pi - e}\).

Step 1: Identify the Type of Series

The given series is \(\sum_{n=0}^{\infty} \left( \frac{e}{\pi} \right)^n\). This is a geometric series where each term is of the form \(a r^n\), with \(a = 1\) and common ratio \(r = \frac{e}{\pi}\).

Step 2: Determine Convergence or Divergence

For a geometric series \(\sum_{n=0}^{\infty} ar^n\) to converge, the absolute value of the common ratio \(|r|\) must be less than 1. Here, \(r = \frac{e}{\pi} \approx 0.865\), since \(\pi > e\), it is true that \(|r| < 1\). Thus, the series converges.

Step 3: Find the Sum of the Convergent Series

The sum \(S\) of a converging geometric series is given by the formula \(S = \frac{a}{1-r}\). With \(a = 1\) and \(r = \frac{e}{\pi}\), the sum is \[ S = \frac{1}{1 - \frac{e}{\pi}} = \frac{\pi}{\pi - e}. \]

Key Concepts

Geometric Series

A geometric series is a series of numbers with a constant ratio between successive terms. Imagine each term in the series being multiplied by the same number to get the next term. This series is defined by its first term 'a' and the common ratio 'r'. For example, in the series \( 1, \, 2, \, 4, \, 8, \, 16, \ldots \), the first term is 1 and the common ratio is 2. Hence, each term is double the preceding one.
To express a geometric series mathematically, it is written as \( \sum_{n=0}^{\infty} ar^n \). This means we start from the term \( a \) and keep multiplying by \( r \) for \( n \) times, where \( n \) ranges from 0 to infinity.
Understanding geometric series is crucial in various fields like finance (interest rates), physics (waves, oscillations), and many more scientific calculations.

• First term \( a \)
• Common ratio \( r \)

Geometric Series Sum Formula

If a geometric series converges, it means that as we keep adding more and more terms, the total sum approaches a certain number. To find this sum, we can use the geometric series sum formula. This formula is given by ˙\( S = \frac{a}{1-r} \).
For the series to converge, the absolute value of the common ratio \(|r|\) should be less than 1. That means the terms get smaller and eventually make the sum stable instead of heading off to infinity.
In practical terms, if you were to have a geometric series with a first term of 5 and a common ratio of 1/2, you'd find the sum of this infinite series as follows:

• First term \( a = 5 \)
• Common ratio \( r = \frac{1}{2} \)
• Sum \( S = \frac{5}{1 - \frac{1}{2}} = 10 \)

This shows how the series, even though infinite, converges to a finite value.

Convergence Criteria

To determine whether a given geometric series converges or diverges, we use the convergence criteria. For geometric series, the key condition is that the absolute value of the common ratio \(|r|\) must be less than 1. If \( |r| < 1 \), the series converges. If \(|r|\) is equal to or greater than 1, the series does not converge; it diverges instead.
Why does this work? Well, if \(|r| \) is less than 1, each successive term is smaller, causing the series to approach a finite sum. This is why the earlier example of the series \(\sum_{n=0}^{\infty} \left( \frac{e}{\pi} \right)^n\) converges—because \(\frac{e}{\pi} \approx 0.865\), making \(|r| < 1\).
Therefore, when dealing with any geometric series, always check \(|r|\) first to conclude whether the series behaves well (converges) or not (diverges).

• Check \(|r| < 1\)
• If true, series converges
• If false, series diverges