Problem 60 - Essential Calculus

Question

Determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty} \frac{1}{4^{n}} $$

Step-by-Step Solution

Answer

The series \( \sum_{n=0}^{\infty} \frac{1}{4^{n}} \) converges and its sum is \( S = \frac{4}{3} \).

Step 1: Identify the Form

Recognize that the series \( \sum_{n=0}^{\infty} \frac{1}{4^{n}} \) is a geometric series, where \( a = 1 \) (the first term when \( n=0 \)) and the common ratio \( r = \frac{1}{4} \).

Step 2: Applying the Convergence Test for Geometric Series

The geometric series converges if and only if the absolute value of \( r \) is less than 1. Here, \( |r| = |\frac{1}{4}| = \frac{1}{4} < 1 \). Thus, this geometric series converges.

Step 3: Find the Sum of the Convergent Series

For a convergent geometric series, the sum can be found using the formula \( S = \frac{a}{1-r} \). Substituting the values of \( a \) and \( r \) into the formula, the sum becomes \( S = \frac{1}{1 - \frac{1}{4}} = \frac{4}{3} \).

Key Concepts

Geometric Series

A geometric series is a special type of series where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio, denoted as \( r \). For instance, the series \( \sum_{n=0}^{\infty} \frac{1}{4^n} \) is geometric because each term is the previous term divided by 4. This is an easy pattern to recognize because, once we know the first term, every subsequent term is predictable using the ratio.
Geometric series are pictorially easy to understand. Imagine you are stacking blocks, with each successive block being a fraction of the previous one (e.g., 1, 1/4, 1/16, etc.). This fixed pattern is a hallmark of geometric series. Recognizing this type of series is crucial, as it determines how we solve for convergence and the sum.

Convergence Test

The convergence of a geometric series depends on its common ratio \( r \). For a geometric series \( \sum_{n=0}^{\infty} ar^n \), it converges if and only if the absolute value of \( r \) is less than 1.

• If \( |r| < 1 \), the series converges, which means the series approaches a finite sum as the number of terms goes to infinity.
• If \( |r| \geq 1 \), the series diverges, meaning it does not approach a specific finite value.

In the original example, since \( r = \frac{1}{4} \) and \(|\frac{1}{4}| < 1\), the series converges. This convergence test mathematically confirms that stacking infinitely many fractions of a block will not lead you to an infinitely large stack, rather it leads to a neat bound.

Sum of Series

Once we've determined that a geometric series converges, finding its sum becomes straightforward thanks to a handy formula: \( S = \frac{a}{1 - r} \). Here, \( a \) is the first term of the series and \( r \) is the common ratio. This formula arises because of the predictable pattern the terms follow, where each term is a fraction of the earlier one.
In our example series, \( a = 1 \) (the value when \( n=0 \)) and \( r = \frac{1}{4} \). Substituting these values into the formula gives us the sum: \( S = \frac{1}{1 - \frac{1}{4}} = \frac{4}{3} \).

• The sum formula is especially useful because it allows us to compute the sum without having to laboriously add each term one by one.
• This makes understanding geometric series useful in various fields, such as financial calculations and physics, where recurring patterns and sums are commonly analyzed.