Discover when adding infinitely many numbers gives a finite answer
Can you add infinitely many numbers and get a finite answer? Surprisingly, yes! Consider this series:
Each term is half the previous one, so the sum gets closer and closer to 2, but never exceeds it. This is called convergence.
But not all infinite series behave this way. If we try to add $1 + 2 + 4 + 8 + \dots$, the sum grows without bound — it diverges.
A geometric series $\sum_{n=0}^{\infty} ar^n$ converges if and only if the absolute value of the common ratio is less than 1:
In other words:
When $|r| < 1$, each term is smaller than the last. As we add more terms, they become negligibly small, so the sum approaches a limit.
When $|r| \geq 1$, terms stay the same size or grow, so the sum keeps growing indefinitely.
When an infinite geometric series converges ($|r| < 1$), its sum is:
This formula comes from the finite series formula $S_n = \frac{a(1 - r^n)}{1 - r}$. As $n \to \infty$, if $|r| < 1$, then $r^n \to 0$, leaving us with:
Experiment with different values to see when series converge or diverge. Watch the partial sums approach (or not approach) a limit!
One of the most elegant applications of infinite geometric series is converting recurring decimals to fractions.
A recurring decimal like 0.333... can be written as an infinite geometric series:
Infinite series appear in physics, engineering, and nature. Here's a classic example:
A ball is dropped from a height. Each bounce reaches a fraction of the previous height. What's the total distance traveled?
Test your understanding with these practice problems on convergence, formulas, and applications.
You've now completed the sequences and series module! You understand arithmetic sequences, geometric sequences, finite series, and infinite series. These concepts form the foundation for calculus and many real-world applications.