Problem 35 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

The sum of the convergent series is 2.

Step 1: Identify the parameters of the series

In this case, the first term \(a\) is 1 and the common ratio \(r\) is \(\frac{1}{2}\).

Step 2: Apply the formula for the sum of infinite geometric series

The formula for the sum of an infinite geometric series is \(\frac{a}{1 - r}\). Substitute the values for \(a\) and \(r\) in the equation.

Step 3: Calculation

After substitution, the equation should look as follows: \(\frac{1}{1 - \frac{1}{2}}\). The calculation simplifies to: \(\frac{1}{\frac{1}{2}} = 2\).

Key Concepts

Convergent Series

A convergent series is a sequence of numbers that has a finite sum when all its terms are added together. This is particularly important in mathematics since it allows for an infinite list of numbers to result in a well-defined, finite number. To determine if a series is convergent, you look for whether the sequence of partial sums tends to a limit, which in this case, is a specific number.

For instance, in the presented exercise, we're looking for the sum of a series where the sequence of numbers continues infinitely. The crucial point is determining whether this infinite sequence converges -- does it have a finite sum? Luckily, there's a specific type of convergent series that makes this process easier to understand: the geometric series.

Geometric Series

A geometric series is a series with a constant ratio between successive terms. Mathematically, a geometric series is represented as \( \sum_{n=0}^{\infty} ar^n \), where \( a \) is the first term and \( r \) is the common ratio. If the absolute value of \( r \) is less than 1, the series is convergent, meaning it has a finite sum given by the formula: \( \frac{a}{1 - r} \).

This formula effectively calculates the sum of all the infinite terms in a geometric series, assuming it's convergent. In the example from the exercise, since the common ratio \( r = \frac{1}{2} \) is less than 1, the series is convergent, and we can confidently use the formula to find the sum.

Infinite Series

An infinite series is a sum of infinitely many terms, which at first thought might appear counterintuitive. After all, how can we talk about the sum of an endless list of numbers? This is where the convergence property of a series becomes critical. If a series is convergent, like the geometric series in our exercise, it can be summed to form a finite number.

Infinite series are classified in various ways depending on their characteristics, such as being arithmetic, geometric, alternating, or various other types based on more complex functions. Geometric series are one of the simplest types of infinite series where terms follow a geometric progression. Understanding whether an infinite series converges or diverges (does not sum to a finite limit) is essential; it dictates whether a finite sum can be found. The concept of summing an infinite series is a cornerstone in fields such as mathematics, physics, engineering, and economics because it allows us to handle concepts of infinity in a practical way.