Problem 82 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$2+\frac{7}{3}+\frac{49}{18}+\frac{343}{108}+\cdots$$

Step-by-Step Solution

Answer

The sum of the given infinite geometric series is -12.

Step 1: Identify the first term and common ratio

Consider the first term (a) from the given infinite geometric series as \(2\). Determine the common ratio (r) by dividing the second term by the first term or third term by the second term. In this case, \(\frac{7}{3}/2 = \frac{49}{18}/\frac{7}{3} = \frac{7}{6}\).

Step 2: Apply infinite geometric series sum formula

If the common ratio's absolute value is less than 1, the sum (S) of the infinite geometric series can be found with the formula \(S = a / (1 - r)\). Substituting the values in, we get – \(S = 2 / (1 - 7/6)\).

Step 3: Solve for S

Calculate S by simplifying the fraction: \(S = 2 / (-1/6) = -12\).

Key Concepts

Sum of Series

In the context of geometry, series refers to the sum of the terms in a sequence. An infinite geometric series, as the name suggests, is a series that continues indefinitely. However, it's fascinating that some infinite series can still have finite sums! That's the ultimate charm of these series.

The infinite geometric series utilizes the sum formula:

• The first term of the series, denoted by \(a\).
• The common ratio, denoted by \(r\), which remains constant between consecutive terms in the series.
• A condition that the absolute value of the common ratio must be less than 1 \((|r| < 1)\), for the series to converge and have a finite sum.

By plugging \(a\) and \(r\) into the formula \(S = \frac{a}{1 - r}\), you can attempt to find the sum of the series. Beware though, if \(|r| \geq 1\), then an infinite series cannot converge to a finite value, making the sum impossible to find. This is essential for determining the feasibility of summing an infinite series.

Common Ratio

The backbone of a geometric series is its common ratio. This is a specific number, \(r\), which defines how each term relates to the next in the series. It's identified by dividing any term in the series by the previous one, providing a constant rate of change.

Let's take a look at the series from the exercise: \(2 + \frac{7}{3} + \frac{49}{18} + \frac{343}{108} + \cdots\). The first term here is \(2\). To find the common ratio, select any two consecutive terms. For example, divide the second term \(\frac{7}{3}\) by the first term \(2\): \[\frac{7}{3} \div 2 = \frac{7}{6}\]The result, \(\frac{7}{6}\), is the common ratio for this series.

Series Convergence

Convergence is a critical property of infinite series, and it determines whether a series can sum to a finite number. In the case of an infinite geometric series, convergence depends on the absolute value of the common ratio \(r\).

A series converges only when

• The absolute value of the common ratio is less than 1, \(|r| < 1\).
• If \(|r| > 1\) or \(|r| = 1\), the series diverges, meaning it does not sum to a finite value.

For our specific exercise, we have determined the common ratio as \(\frac{7}{6}\). Notably, \(\left|\frac{7}{6}\right| = \frac{7}{6} > 1\), which means the series fails to converge. Without convergence, the sum cannot be finite, which is why calculating the sum of this particular series appears impossible.

Mathematical Proof

Proofs provide robust verification of mathematical concepts and calculations. In the scenario of our specific infinite geometric series, a proof helps affirm that the series cannot converge due to the common ratio.

Using the series formula for the sum \(S = \frac{a}{1 - r}\), proves vital:

• Substitute \(a = 2\) and \(r = \frac{7}{6}\).
• Calculate \(S = \frac{2}{1 - \frac{7}{6}} = \frac{2}{-\frac{1}{6}} = -12\).
• The negative result arises from the attempt to apply the formula where it should not, since \(|r| > 1\).

Mathematically, it reveals the inapplicability of the formula due to the non-convergent nature (\(r \geq 1\)), hence the sum finding is incorrect under these conditions. This proof highlights the importance of verifying conditions before slimming down to calculations, avoiding invalid results.