Convergence of Series
“Adding forever can still give a finite answer—as long as the pieces shrink fast enough.”
The formula
S = Σ(n=1..∞) aₙ = lim(N→∞) (a₁ + a₂ + ⋯ + a_N); Σ(n=0..∞) arⁿ = a/(1−r), |r|<1How to read it: the sum of infinitely many terms is defined as the limit approached by the partial sums of the first N terms
- aₙ
- — the n-th term of the series
- Σ(n=1..N)
- — the finite sum of just the first N terms
- S
- — the limit the partial sums approach—the sum of the series
- r
- — the common ratio of a geometric series—converges when |r|<1
The hook
You can add infinitely many numbers and still get a finite answer—1 + ½ + ¼ + ⅛ + ⋯ = 2.
In plain words
The 'sum' of an infinite series is not literally added to the end. You form the partial sum of the first N terms; if that value approaches some number S as N grows, the series 'converges to S'.
The intuition
Suppose you cross a room by covering half the remaining distance each step: ½, ¾, ⅞, … You never quite reach the wall, yet the total distance approaches 1 (one room). Infinitely many pieces fit inside a finite wall.
How it's built
What matters is how fast the terms aₙ shrink to 0. If they halve each time (ratio ½), the partial sums close in on a wall and converge. If they shrink too slowly, the partial sums grow without bound and diverge. A geometric series settles exactly to a/(1−r) when |r|<1.
Example
Σ(n=0..∞)(½)ⁿ = 1 + ½ + ¼ + ⅛ + ⋯. The partial sums are 1, 1.5, 1.75, 1.875, … → 2. By the formula, a=1, r=½ gives 1/(1−½)=2.
Common mistake
Does terms going to 0 guarantee convergence? No. The harmonic series 1 + ½ + ⅓ + ¼ + ⋯ has terms going to 0 yet diverges to infinity. Going to 0 is not the same as going to 0 fast enough.
Where it's used
Infinite decimal expansions, present value of annuities, approximating functions by Taylor series, error analysis in iterative computation—series convergence underlies every place infinity is handled finitely.
Where it came from
Zeno's paradox (Achilles and the tortoise) puzzled people for over two thousand years, until Cauchy made 'infinite sums' rigorous with the definition as a limit of partial sums.
Prerequisites
Quick check
What is Σ(n=0..∞)(1/3)ⁿ = 1 + 1/3 + 1/9 + ⋯ ?
- 3/2✓
- 1/2
- 3
- it diverges
Practice
Find the sum Σ(n=0..∞)(1/2)ⁿ = 1 + 1/2 + 1/4 + ⋯.
Answer: 2
- geometric series a=1, r=1/2
- a/(1−r) = 1/(1−1/2) = 2
Takeaway: A geometric series is just a/(1−r).
Find the sum Σ(n=1..∞)(1/2)ⁿ = 1/2 + 1/4 + 1/8 + ⋯.
Answer: 1
- first term a=1/2, ratio r=1/2
- a/(1−r) = (1/2)/(1/2) = 1
Takeaway: Where the sum starts changes the first term a.
Find the sum Σ(n=0..∞)(0.9)ⁿ.
Answer: 10
- a=1, r=0.9
- 1/(1−0.9) = 1/0.1 = 10
Takeaway: The closer r is to 1, the larger the sum explodes.
Does the harmonic series 1 + 1/2 + 1/3 + 1/4 + ⋯ converge or diverge?
Answer: 1
- the terms 1/n go to 0, but too slowly
- the partial sums grow without bound, so it diverges
Takeaway: Terms going to 0 can still diverge—the harmonic series proves it.