The question "How do you derive the infinite series?" is somewhat incomplete; it's more accurate to ask how to express or represent an infinite series, or, given a sequence of numbers, how to determine the general expression for the sum of the infinite sequence. Based on the provided reference, we'll focus on how to express an infinite geometric series.
Expressing an Infinite Geometric Series
An infinite geometric series is a series where each term is multiplied by a constant value (the common ratio) to get the next term. To derive, meaning to express this infinite geometric series, we use the following formula:
The Formula for Infinite Geometric Series
The infinite series formula for a geometric series is:
∑ k = 1 ∞ a r k − 1
Where:
- ∑ (Sigma) denotes the sum of all terms.
- k is an index (counter) that begins at 1 and goes to infinity (∞).
- a represents the first term of the series.
- r is the common ratio. Each term is the previous term multiplied by the common ratio.
- k-1 exponent signifies that the first term is when k=1; Thus, the first term is ar^(1-1) = ar^0 = a. The second term is when k=2, Thus, the second term is ar^(2-1) = ar^1 = ar.
How to Use the Formula
- Identify the first term (a): This is the starting number of the series.
- Determine the common ratio (r): This is the value you multiply by to get from one term to the next.
- Plug 'a' and 'r' into the formula: This results in the general expression for the infinite geometric series.
Example
Let's say you have the geometric series: 2 + 1 + 1/2 + 1/4 + ...
- First term (a) = 2
- Common ratio (r) = 1/2 (each term is half the previous term).
So, the infinite series would be expressed as:
∑ k = 1 ∞ 2 (1/2) k − 1
Finding a Specific Term
To find the nth term of this series, you would simply replace k with n in the formula immediately after the sigma symbol. For example, the third term (n=3) of the series, would be:
2(1/2)^(3-1) = 2(1/2)^2 = 2*(1/4) = 1/2
Convergence
It's important to note that an infinite geometric series only has a finite sum (converges) if the absolute value of the common ratio (|r|) is less than 1. Otherwise, the series diverges, meaning its sum is infinite. For our above example |1/2| < 1, so this series converges.
Summary
| Step | Action | Example |
|---|---|---|
| 1. Identify 'a' | Find the first term of the sequence | In the series 2 + 1 + 1/2 + ..., a = 2 |
| 2. Identify 'r' | Find the common ratio between each term | In the series 2 + 1 + 1/2 + ..., r = 1/2 |
| 3. Express the series | Use the formula ∑ k = 1 ∞ a r k − 1 | ∑ k = 1 ∞ 2 (1/2) k − 1 |
| 4. Find a specific term | replace k with the term number 'n' in the a r k − 1 equation | To find the 3rd term, 2*(1/2)^(3-1) = 1/2 |
