Infinite Geometric Sequences and Their Sums
Infinite Convergent Geometric Sequences
An infinite geometric sequence is geometric sequence that continues indefinitely.
One is considered convergent if its terms approach a finite limit as the number of terms increases indefinitely.
Example
- Let's consider an infinite geometric sequence with $u_1 = 1$ and $r = \frac12$.
- The terms of this series go: $1, \frac12, \frac14, \frac18, \frac16...$.
- Eventually, as the terms go to infinity, each term gets closer and closer to 0, so we say it is convergent.
Condition for Convergence
- Whether an infinite geometric sequence converges depends on the value of its common ratio.
- The condition for convergence is: $$|r|< 1$$ where $|r|$ represents the absolute value (or modulus) of the common ratio.
Hint
- To quickly determine if a sequence converges, check if the absolute value of the common ratio is less than 1.
- If it is, the sequence converges; if not, it diverges.
Formula for the Sum to Infinity
- The sum of a finite geometric series is given by: $$S_n = \frac{u_1(1-r^n)}{1-r}
- If $|r| < 1$, as $n$ approaches $\infty$, $r^n$ approaches $0$.
- Thus, the formula for the sum of an infinite convergent geometric series is: $$S_{\infty} = \frac{u_1}{1-r}$$
Note
- This formula only applies when $|r| < 1$.
- If $|r| \geq 1$, the sequence diverges and the sum to infinity does not exist
Exam technique
Applying the Formula
To apply this formula, follow these steps:
- Identify the first term $a$ and the common ratio $r$.
- Check if $|r|< 1$.
- If not, the sum to infinity doesn't exist.
- If $|r| < 1$, substitute $a$ and $r$ into the formula $S_{\infty} = \frac{a}{1-r}$.
