Problem 60 - Algebra and Trigonometry

Question

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$ -\frac{100}{9}+\frac{10}{3}-1+\frac{3}{10}-\cdots $$

Step-by-Step Solution

Answer

The series is convergent and its sum is \(-\frac{1000}{81}\).

Step 1: Determine the first term

Identify the first term of the given infinite geometric series. The first term is given as \(-\frac{100}{9}\).

Step 2: Determine the common ratio

To find the common ratio \(r\), divide the second term by the first term: \(r = \frac{ \frac{10}{3} }{ -\frac{100}{9} } = -\frac{10}{3} \cdot -\frac{9}{100} = \frac{30}{300} = \frac{1}{10} \).

Step 3: Determine convergence

A geometric series converges if the absolute value of the common ratio \(|r|\) is less than 1. In this case, \(|\frac{1}{10}| = 0.1\), which is less than 1, so the series converges.

Step 4: Find the sum of the convergent series

For a convergent infinite geometric series with first term \(a\) and common ratio \(r\), the sum \(S\) is calculated using the formula \(S = \frac{a}{1 - r}\). Substitute the values to find the sum: \(S = \frac{-\frac{100}{9}}{1 - \frac{1}{10}} = \frac{-\frac{100}{9}}{\frac{9}{10}} = -\frac{100}{9} \cdot \frac{10}{9} = -\frac{1000}{81}\).

Key Concepts

Convergence

In mathematics, **convergence** refers to the idea of a sequence or series approaching some limit as more terms are added. Imagine taking steps that are successively smaller but always in the right direction - you will eventually land on a target point.

For an infinite geometric series, whether it converges (comes to a particular finite value) or diverges (grows uncontrollably) is centred on the series' common ratio. A geometric series will converge if the absolute value of the common ratio \( |r| \) is less than 1.

This is because as you multiply the terms by a number smaller than 1 repeatedly (in absolute value), the terms themselves keep getting smaller and smaller, cumulatively approaching a particular value. In this example, since the common ratio is \( |\frac{1}{10}| = 0.1 \), which is clearly less than 1, the series converges.

Common Ratio

The **common ratio** is the constant factor between consecutive terms of a geometric sequence. It essentially tells us how each term relates to the previous term by multiplication.

For our infinite series, the common ratio can be found by dividing any term in the series by the previous one.

• Here, the second term is \( \frac{10}{3} \) and the first term is \(-\frac{100}{9}\).
• So, the common ratio \( r \) is calculated as \( \frac{\frac{10}{3}}{-\frac{100}{9}} \). This simplifies down to \( \frac{1}{10} \).

The sign and size of the common ratio are vital as:

• If \( |r| < 1 \), the terms keep getting smaller —leading to convergence.
• If \( |r| \geq 1 \), the series diverges.

In our example, because \( |\frac{1}{10}| = 0.1 \), which is less than 1, the series converges.

Infinite Series

An **infinite series** extends indefinitely beyond a starting point. It consists of the sum of an infinite sequence of numbers. The presence of infinity opens doors to exciting phenomena in mathematics, such as convergence and divergence.

Infinite series is denoted with a sum notation, often written with an ellipsis \( \cdots \) hinting its continuation indefinitely. In our example, we have an infinite series beginning with \(-\frac{100}{9}+\frac{10}{3}-1+\frac{3}{10}-\cdots\).

• Mathematicians seek specific characteristics, such as the common ratio, to determine the series' behaviour.
• Determining whether an infinite series converges or diverges can unravel insights about the nature of its terms as we move towards infinity.

In the given problem, the presence of a convergent series due to a common ratio (\(|\frac{1}{10}|\)) less than 1 assures us that the infinite series reaches a finite sum.

Sum of Series

Finding the **sum of a series** means calculating the total value of all terms in the series combined. For finite series, it's straightforward: just add the terms. However, infinite series require more advanced techniques.

For a **convergent infinite geometric series**, there is a specific formula to find its sum:

• Sum \( S = \frac{a}{1-r}\)
• Here, \( a \) is the first term of the series, and \( r \) is the common ratio.

With our example, the sum is determined using the formula: \( S = \frac{-\frac{100}{9}}{1-\frac{1}{10}} = -\frac{1000}{81}\). It reveals that despite having infinite terms, their contributions eventually converge to a finite number, \(-\frac{1000}{81}\), thanks to the shrinking nature of the terms influenced by the \( 0.1 \) common ratio.