Problem 50 - 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. $$ \sum_{n=0}^{\infty}(\sqrt{2})^{n} $$

Step-by-Step Solution

Answer

The series diverges because the common ratio \( \sqrt{2} > 1\).

Step 1: Recognize the series

This is a geometric series of the form \( \sum_{n=0}^{\infty} ar^n \), where \( a = 1 \) (the first term) and \( r = \sqrt{2} \) (the common ratio).

Step 2: Check the convergence criterion for geometric series

A geometric series converges if the absolute value of the common ratio, \( |r| \), is less than 1. Here \( r = \sqrt{2} \) and \( |\sqrt{2}| = \sqrt{2} \approx 1.414 \). Since \( \sqrt{2} > 1 \), the series does not meet the convergence criterion.

Step 3: Determine the convergence

Since \( |r| = \sqrt{2} > 1 \), the series \( \sum_{n=0}^{\infty} (\sqrt{2})^n \) is divergent. Geometric series with \( |r| \geq 1 \) diverge.

Key Concepts

Convergence of Series

To understand whether a series converges or diverges, it's important to know what convergence means. In simple terms, when we say a series converges, we mean the sum of its terms approaches a particular finite number as we keep adding more terms. Think of convergence like money going into a savings account: as you keep adding small amounts, the total grows towards a specific goal.

An infinite series like \( \sum_{n=0}^{\infty} a_n \) converges if the sequence of partial sums \( S_N = a_0 + a_1 + a_2 + \ldots + a_N \) approaches a finite limit as \( N \to \infty \).

• The series adds up to a finite number.
• If this finite sum exists, the series is convergent.
• Such series are predictable and stable in behavior.

Convergence is central in many mathematical applications because it allows for approximations of numbers like \( \pi \) or \( e \), and in solving differential equations.

Series Divergence

In contrast to convergence, series divergence occurs when the sum of the series does not settle towards any finite number. Imagine trying to save for something but your savings never reach the target, no matter how much you add. That's what divergence is like in mathematics.

For the series \( \sum_{n=0}^{\infty} a_n \) to diverge, the sequence of partial sums \( S_N = a_0 + a_1 + a_2 + \ldots + a_N \) must not approach a finite limit as \( N \to \infty \). This could mean the sums grow larger and larger without bound, or they simply fluctuate without approaching a specific value.

• Divergent series do not sum to a finite number.
• They can grow indefinitely or behave erratically.

Knowing whether a series diverges is vital as it helps avoid futile calculations and ensures accurate predictions or interpretations in practical applications.

Geometric Series Criterion

The geometric series criterion provides a straightforward way to determine convergence or divergence specifically for geometric series. A geometric series is characterized by each term being a constant multiple of the previous term. This series takes the form:\[ \sum_{n=0}^{\infty} ar^n \]where \( a \) is the initial term and \( r \) is the common ratio.

The convergence of a geometric series largely depends on the common ratio \( r \).

• If \( |r| < 1 \), the series converges, and its sum is \( \frac{a}{1-r} \).
• If \( |r| \geq 1 \), the series diverges.

Applying this to our example, the series with \( r = \sqrt{2} \approx 1.414 \) clearly diverges, since the absolute value of the common ratio exceeds 1. Geometric series provide a clear and quick decision rule, greatly simplifying many problems in calculus and engineering.