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.
For 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.
TipTo 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}$$
NoteThis formula only applies when $|r| < 1$. If $|r| \geq 1$, the sequence diverges and the sum to infinity does not exist.
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 $S_{\infty} = \frac{a}{1-r}$.
