Problem 43 - Essential Calculus

Question

Find the sum of the convergent series. $$ \sum_{n=0}^{\infty}\left(\frac{1}{2^{n}}-\frac{1}{3^{n}}\right) $$

Step-by-Step Solution

Answer

The sum of the inconvergent series is \( \frac{1}{2} \).

Step 1: Identify the Terms and Calculate the Sum for each Series

First we can recognize that the series is a sum of two geometric series. We separate them into two parts and get \( \sum_{n=0}^{\infty}\frac{1}{2^{n}} \) and \( \sum_{n=0}^{\infty}\frac{1}{3^{n}} \). For each series, we can use the sum formula for infinite geometric series \[ S = \frac{a}{1-r} \], where \( a \) is the first term of the series and \( r \) is the common ratio.

Step 2: Calculate and Subtract the Sums

For the sum of the first geometric series, we get: \( S_1 = \frac{1}{1 - \frac{1}{2}} = 2 \). For the sum of the second geometric series, we get: \( S_2 = \frac{1}{1 - \frac{1}{3}} = \frac{3}{2} \). Finally, the result of the whole series is obtained by subtracting the sum of the second geometric series from the sum of the first geometric series, \( S = S_1 - S_2 = 2 - \frac{3}{2} \).

Step 3: Simplify the Result

This will yield the final result of the problem. Simplify the result to get: \( S = \frac{1}{2} \).

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term, after the first, is found by multiplying the previous term by a fixed non-zero number called the common ratio, denoted as \( r \). For instance, in the geometric series \( 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots \), each number is half of the one before it.

• **First Term**: The first term of the series, often represented with \( a \).
• **Common Ratio**: The quotient of any term and the term preceding it.

Geometric series are incredibly useful in various fields, such as economics and physics, for their simplicity in calculating the sum of terms over an infinite range when the common ratio is less than 1 in absolute value.

Infinite Series

An infinite series is simply the sum of terms taken from a sequence in an endless manner. Unlike finite series, where the sequence stops, infinite series continue indefinitely.

Infinite series can be daunting due to their non-terminating nature but are highly significant in advanced mathematics and physics. The example series given in the exercise includes terms like \( \frac{1}{2^n} \) and \( \frac{1}{3^n} \), ongoing endlessly as \( n \) approaches infinity.

• **Convergence**: A series converges if the sum of its terms approaches a finite limit.
• **Divergence**: A series diverges if the total sum tends to infinity.

The key value of infinite series is in convergence, where they can represent real numbers or functions when summed appropriately.

Series Sum Formula

The series sum formula for an infinite geometric series provides a simple way to find the sum when the sequence is convergent. The steady formula is given by \( S = \frac{a}{1-r} \), where \( S \) is the sum, \( a \) is the first term, and \( r \) is the common ratio. This formula only applies when the absolute value of the common ratio \( |r| < 1 \).

Using this formula greatly simplifies calculations, as seen in the exercise, where two geometric series were evaluated:

• The first series with a first term \( a = 1 \) and a common ratio \( r = \frac{1}{2} \), yielded a sum of \( S = 2 \).
• The second series with \( a = 1 \), and \( r = \frac{1}{3} \), resulted in a sum of \( S = \frac{3}{2} \).

Finally, subtracting these sums, due to the nature of the original series as a difference between two series, provided the final convergent sum of \( \frac{1}{2} \). Understanding the series sum formula is crucial for breaking down infinite geometric series into manageable parts and calculating exact sums with ease.