Problem 120 - Precalculus with Limits

Question

Exercises 117-120, find the sum of the infinite series. \( \displaystyle \sum_{i=1}^{\infty} 2 \left(\frac{1}{10} \right)^i \)

Step-by-Step Solution

Answer

The sum of the given infinite series is \( \frac{2}{9}\) .

Step 1: Verification

The series given by the exercise is \( \displaystyle \sum_{i=1}^{\infty} 2 \left(\frac{1}{10} \right)^i \). We can rewrite this as \(2 \displaystyle \sum_{i=1}^{\infty} \left(\frac{1}{10} \right)^i\). Thus the series is written in the basic form for a geometric series which is \(a*r^{n-1}\), where a is the first term, r is the ratio, and n is the term number. In this case, a = 1/5 , r = 1/10.

Step 2: Conditions

Before we proceed with the calculation we need to ensure that the condition for the convergence of the geometric series is respected. In other words, the absolute value of r should be less than 1. In this case, |r| = |1/10| < 1. Hence, the condition is respected and we can proceed with the calculation of the sum.

Step 3: Calculation

To find the sum of a convergent infinite geometric series, we use the formula: S= \frac{a}{1-r}. Substituting the values of a and r, we obtain S= \(\frac{1}{5}\) / (1 - \( \frac{1}{10}\)) = \( \frac{1}{5}\) / \( \frac{9}{10} = \frac{2}{9}\)

Key Concepts

Geometric Series Explained

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. It's a way to represent repeated multiplication through addition.
In the provided exercise, the series is \( \sum_{i=1}^{\infty} 2 \left(\frac{1}{10} \right)^i \). Notice how each term is generated by multiplying the previous term by \( \frac{1}{10} \), which is the common ratio here.

A geometric series can be finite or infinite. An infinite series continues indefinitely, but remarkably, it can still have a sum, provided certain conditions are met.

• The first term in a geometric series is usually denoted by \( a \).
• The common ratio is denoted by \( r \).
• In our example, \( a = 2 \times \frac{1}{10} = \frac{1}{5} \) and \( r = \frac{1}{10} \).

A geometric series is the perfect example of how multiplication and repeated patterns simplify complex mathematical phenomena.

Understanding Convergence

Convergence of a series is a key concept in calculus and mathematical analysis. It tells us whether an infinite series results in a finite sum or not. For a geometric series to converge, the absolute value of the common ratio \( r \) must be less than 1.

In mathematical terms, if \(|r| < 1\), the series converges; if \(|r| \geq 1\), it diverges and doesn't have a finite sum. This is because when \(|r| < 1\), the successive terms in the series become smaller and smaller, approaching zero.
For the given series, \( r = \frac{1}{10} \), and clearly \(|\frac{1}{10}| < 1\). This means our series is convergent.

• Convergence allows us to calculate the sum of the series even if it includes infinite terms.
• This is essential in various fields of science and engineering where infinite processes are often modeled.

By understanding convergence, one can approach complex infinite series with confidence, knowing when an actual sum can be calculated.

Calculating the Sum of Series

To find the sum of an infinite geometric series that converges, we use the formula:
\[S = \frac{a}{1 - r}\]
where \( S \) is the sum, \( a \) is the first term, and \( r \) is the common ratio.

For the series \( \sum_{i=1}^{\infty} 2 \left(\frac{1}{10} \right)^i \), we have \( a = \frac{1}{5} \) and \( r = \frac{1}{10} \). Plugging these values into the formula gives:
\[S = \frac{\frac{1}{5}}{1 - \frac{1}{10}} = \frac{\frac{1}{5}}{\frac{9}{10}}\]
Simplifying this expression:
\[S = \frac{1}{5} \times \frac{10}{9} = \frac{2}{9}\]
So, the infinite series sums up to \( \frac{2}{9} \).

• This result illustrates the power of using the geometric series sum formula.
• It shows how infinite sequences can be managed and understood using mathematical principles.

It's fascinating how you can add up infinite parts to get a finite number, reinforcing the beauty and consistency of mathematics.