A geometric series is the sum of the terms of a geometric sequence, where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This series can be expressed as $$S_n = a + ar + ar^2 + ar^3 + ... + ar^{n-1}$$, where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms. Geometric series can be finite or infinite, and their behavior varies significantly based on the value of the common ratio.
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The formula for the sum of a finite geometric series is $$S_n = a \frac{1 - r^n}{1 - r}$$ when the common ratio 'r' is not equal to 1.
For an infinite geometric series, if the absolute value of the common ratio 'r' is less than 1, the series converges to $$S = \frac{a}{1 - r}$$.
If 'r' is greater than or equal to 1 for an infinite series, it diverges, meaning it does not have a finite sum.
Geometric series can model real-life situations like population growth, interest calculations, and certain financial scenarios due to their multiplicative nature.
Understanding geometric series is crucial for calculating present and future values in finance, particularly when dealing with loans and investments.
Review Questions
How does the common ratio affect the behavior of a geometric series, particularly in terms of convergence or divergence?
The common ratio plays a critical role in determining whether a geometric series converges or diverges. If the absolute value of the common ratio 'r' is less than 1, the series converges to a finite sum. Conversely, if 'r' is greater than or equal to 1, the series diverges, resulting in an infinite sum that does not stabilize. This understanding helps in identifying when a series can yield meaningful results in applications like finance or natural phenomena.
Derive the formula for the sum of a finite geometric series and explain its components.
To derive the formula for a finite geometric series, start with $$S_n = a + ar + ar^2 + ... + ar^{n-1}$$. By multiplying both sides by 'r', we get $$rS_n = ar + ar^2 + ... + ar^n$$. Subtracting these two equations leads to $$S_n - rS_n = a - ar^n$$, which simplifies to $$S_n(1 - r) = a(1 - r^n)$$. Finally, dividing by $(1 - r)$ gives us $$S_n = a \frac{1 - r^n}{1 - r}$$. Here, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.
Evaluate how knowledge of geometric series can be applied in real-world scenarios, particularly in finance.
Knowledge of geometric series is essential in finance for calculating present and future values of investments and loans. For instance, when assessing compound interest, which grows exponentially over time, using geometric series allows us to find total amounts after several periods. The formula for an infinite geometric series also applies in calculating perpetuities in annuities. By understanding these applications, individuals can make informed decisions about savings and investments.
Related terms
Geometric Sequence: A sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio.
Common Ratio: The fixed number that each term in a geometric sequence is multiplied by to obtain the next term.
Summation Notation: A concise way to represent the sum of a sequence of numbers, often using the sigma symbol (Σ) to indicate the summation of terms.