Problem 47 - Algebra 2

Question

Technology Create a spreadsheet to evaluate the first \(n\) terms of each series. Determine whether each infinite series converges to a sum. If so, estimate the sum. $$ \sum_{n=1}^{\infty} \frac{1}{2^{n}} $$

Step-by-Step Solution

Answer

The series \(\sum_{n=1}^{\infty} \frac{1}{2^{n}}\) converges and the sum is 1.

Step 1: Identify the terms of the series

For the given series \(\sum_{n=1}^{\infty} \frac{1}{2^{n}}\), we can identify that the first term 'a' is \(\frac{1}{2}\) (when n=1), and the common ratio 'r' is also \(\frac{1}{2}\).

Step 2: Determine the convergence of the series

A geometric series \(\sum_{n=1}^{\infty} ar^{n-1}\) converges if the absolute value of the common ratio is less than 1, i.e., |r| < 1. In this case, the common ratio is \(\frac{1}{2}\), thus the series converges.

Step 3: Calculate the sum of the series

We can calculate the sum of the series using the formula \(S = \frac{a}{1 - r}\). Substituting the values \(a = \frac{1}{2}\) and \(r = \frac{1}{2}\) we get, \(S = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = 1\)

Key Concepts

Convergence

In mathematics, convergence plays a crucial role in understanding whether an infinite series will add up to a finite number. For geometric series, convergence is determined by the value of the common ratio, denoted as 'r'. A geometric series will converge if the absolute value of its common ratio is less than 1. This means \(|r| < 1\) is the key condition.

In the given exercise, where the infinite series is \( \sum_{n=1}^{\infty} \frac{1}{2^{n}} \), the common ratio \(r\) is \(\frac{1}{2}\). Since \(\frac{1}{2} < 1\), this series converges. Convergence ensures that as we keep adding more terms, the series approaches a finite limit rather than growing indefinitely.

Sum of Series

Once it's established that an infinite geometric series converges, the next step is finding its sum. The formula to calculate the sum \(S\) of a converging geometric series \(\sum_{n=0}^{\infty} ar^{n}\) is \(S = \frac{a}{1 - r}\). Here, 'a' denotes the first term of the series, while 'r' is the common ratio.

For the series in the exercise, \(a = \frac{1}{2}\) and \(r = \frac{1}{2}\). Plug these into the formula, and you get

• \(S = \frac{\frac{1}{2}}{1 - \frac{1}{2}}\)
• \(S = \frac{\frac{1}{2}}{\frac{1}{2}}\)
• \(S = 1\)

This calculation shows that the infinite series sums to a precise value of 1, which is the estimated sum of our infinite series.

Common Ratio

The common ratio 'r' is a fundamental concept in geometric series. It is the number by which each term is multiplied to get the next term. Identifying the common ratio is essential for determining the series behavior.

For example:

• In the series \( \sum_{n=1}^{\infty} \frac{1}{2^{n}} \), the common ratio is \(\frac{1}{2}\).
• This means each term is half of the previous term.

Understanding and correctly identifying the common ratio allows us to apply convergence tests and find the sum effectively. As highlighted in the exercise, if the ratio had been greater than or equal to 1, the series would not converge to a finite limit.

Infinite Series

An infinite series, in simple terms, is the sum of an infinite sequence of numbers. These types of series are intriguing because they extend indefinitely, yet they can still reach a finite sum under certain conditions.

The infinite series \( \sum_{n=1}^{\infty} \frac{1}{2^{n}} \) consists of terms that get progressively smaller. The infinite nature of this series does not prevent it from having a sum, as it satisfies the convergence criteria.

Infinite series are a fundamental concept in calculus and analysis, allowing mathematicians to work with and derive results for endless sums. By understanding whether an infinite series converges and knowing how to calculate its sum, one can tackle a wide range of mathematical problems effectively.