Lesson 5.3: Geometric Series and the Sum to Infinity
Introduction
In this lesson, we will explore geometric series, a fundamental concept in mathematics, especially in the context of finance. By understanding how geometric series work, you will be able to analyze various financial scenarios involving growth and decay, such as savings accounts and loans. Our objectives for this lesson are:
- Understand the nth term and the sum of a geometric series.
- Learn how to find the sum to infinity for a convergent geometric series and the condition for this convergence.
- Recognize examples of geometric growth and decay in real-world situations.
- Calculate the nth term and the sum of a geometric series.
- Find the sum to infinity and state the condition necessary for convergence.
What is a Geometric Series?
Definition
A geometric series is the sum of the terms of a geometric sequence. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio, denoted as $ r $.
The $ n $-th term of a geometric sequence can be expressed as:
$$
$t_n = a \cdot r^{n-1}$
$$
where:
- $ t_n $ is the $ n $-th term,
- $ a $ is the first term, and
- $ r $ is the common ratio.
General Formula for the Sum of the First $ n $ Terms
The sum $ S_n $ of the first $ n $ terms of a geometric series can be calculated using the formula:
$$
$S_n = a \frac{1-r^n}{1-r} \quad (r $
eq 1)
$$
This formula allows us to sum the first $ n $ terms of a geometric series quickly.
Example Calculation
Example: Calculate the sum of the first 5 terms of a geometric series where $ a = 3 $ and $ r = 2 $.
- Identify the first term ($ a $) and the common ratio ($ r $):
- $ a = 3 $
- $ r = 2 $
- Plug these values into the formula for $ S_n $:
$$
S_5 = $3 \frac{1-2^5}{1-2}$ = $3 \frac{1-32}{-1}$ = $3 \cdot 31$ = 93
$$
So, the sum of the first 5 terms is 93.
The Sum to Infinity
Definition
Not all geometric series converge. A geometric series converges to a finite limit if the absolute value of the common ratio $ |r| < 1 $. In such cases, we can determine the sum to infinity:
$$
$S_{\infty} = \frac{a}{1-r}$
$$
This formula provides the sum of an infinite geometric series where $ |r| < 1 $.
Example of Convergence
Example: A bank account offers a savings plan where the first deposit is $100 and earns an interest rate of 5% per year. The interest is compounded annually. Determine the total amount in the account after an infinite number of years assuming the interest keeps getting added.
- Identify the first term and common ratio:
- $ a = 100 $
- $ r = 1 + 0.05 = 1.05 $
- Check the condition for convergence:
- Since $ |1.05| > 1 $, this series does not converge. Therefore, the total amount will grow indefinitely.
- However, for a case of decay such as depreciation or if the interest rate were negative, you might find a sum to infinity. For an example with decay:
- Suppose you receive a return of 2% per year instead:
- $ a = 100 $
- $ r = 1 - 0.02 = 0.98 $
- Now, calculate $ S_{\infty} $:
$$
S_{$\infty$} = $\frac{100}{1 - 0.98}$ = $\frac{100}{0.02}$ = 5000
$$
Thus, if the interest rate were less than the financial gain, we could calculate it as converging to a finite limit of 5000.
Recognizing Geometric Growth and Decay
In many real-world scenarios, geometric growth can be seen in finance. For example, investments and savings grow exponentially because of compound interest. Conversely, decay can be seen in depreciation of assets.
When analyzing such situations, remember:
- Growth occurs when $ |r| > 1 $ (the series diverges).
- Decay occurs when $ |r| < 1 $ (the series converges).
Finding the nth Term and Sum of a Geometric Series
Examples
- Finding the nth term: For a geometric series where $ a = 5 $ and $ r = 3 $, find the 4th term.
- Use the nth term formula:
$$
t_4 = $5 \cdot 3^{4-1}$ = $5 \cdot 27$ = 135
$$
Thus, the 4th term is 135.
- Finding the sum: If we want to sum the first 4 terms of the same series:
- Calculate $ S_4 $:
$$
S_4 = $5 \frac{1-3^4}{1-3}$ = $5 \frac{1-81}{-2}$ = $5 \cdot 40$ = 200
$$
Therefore, the sum of the first 4 terms is 200.
Conclusion
In this lesson, we learned about geometric series and how to calculate their sums and the conditions for convergence. Understanding these concepts is critical in various applications, particularly in financial mathematics. The skills you gained will be invaluable when analyzing financial scenarios and understanding the behavior of geometric sequences.
Study Notes
- A geometric series is defined by a first term $ a $ and a common ratio $ r $.
- The nth term of a geometric series is given by $ t_n = a \cdot r^{n-1} $.
- The sum of the first $ n $ terms is calculated using $ S_n = a \frac{1 - r^n}{1 - r} $ for r
eq 1 .
- The sum to infinity, if $ |r| < 1 $, is given by $ S_{\infty} = \frac{a}{1 - r} $.
- Recognize situations of growth (where $ |r| > 1 $) and decay (where $ |r| < 1 $).