An infinite series is a way to add the terms of a sequence forever. Instead of asking only where individual terms go, calculus asks what happens to the running total as more and more terms are added. This idea matters because many functions, measurements, and physical processes can be represented as sums of infinitely many simpler pieces.
Infinite series connect algebra, limits, approximation, and real-world modeling.
The key object is the partial sum, which is the sum of the first n terms of a series. If the partial sums approach a finite number as n grows without bound, the series converges to that number. If the partial sums do not approach a finite number, the series diverges.
A sequence is a list of terms, while a series is the sum formed from those terms.
Key Facts
- A sequence is written a1, a2, a3, ... and lists terms in order.
- An infinite series is written a1 + a2 + a3 + ... or sum from k = 1 to infinity of ak.
- The nth partial sum is Sn = a1 + a2 + ... + an.
- A series converges if lim n to infinity Sn = S for some finite number S.
- A necessary condition for convergence is lim n to infinity an = 0.
- Geometric series formula: a + ar + ar^2 + ... = a/(1 - r) when |r| < 1.
Vocabulary
- Sequence
- A sequence is an ordered list of numbers, usually written as a1, a2, a3, and so on.
- Series
- A series is the sum of the terms of a sequence.
- Partial Sum
- A partial sum is the sum of a finite number of terms from a series.
- Convergence
- Convergence means the partial sums of an infinite series approach a finite limiting value.
- Divergence
- Divergence means the partial sums of an infinite series do not approach a finite limiting value.
Common Mistakes to Avoid
- Confusing a sequence with a series. A sequence lists terms, while a series adds those terms.
- Assuming that an approaching 0 guarantees convergence. The terms must approach 0 for convergence, but that condition alone is not enough.
- Adding only a few terms and declaring the exact sum. A finite partial sum may be a good approximation, but the infinite series equals the limit of all partial sums.
- Using the geometric series formula when |r| is not less than 1. The formula a/(1 - r) applies only when the common ratio has absolute value less than 1.
Practice Questions
- 1 Find the first four partial sums of the series 1/2 + 1/4 + 1/8 + 1/16 + ... . What value do the partial sums appear to approach?
- 2 Determine whether the geometric series 3 + 1.5 + 0.75 + 0.375 + ... converges, and if it does, find its sum.
- 3 A sequence has terms an = 1/n. Explain why the fact that an approaches 0 does not by itself prove that the series 1 + 1/2 + 1/3 + 1/4 + ... converges.