Equations · Algebra and geometry
Geometric series
The sum of the endless series 1 + r + r² + r³ + …, when r is between −1 and 1.
∑ₖ₌₀^∞ rᵏ = 1 / (1 − r)
Aethel says
the sum from k equals 0 to infinity of r to the power of k, equals 1 over the quantity 1 minus r
What it says
Each term is the previous one multiplied by the same ratio r. When r is smaller than 1 in size, the terms shrink fast enough that the infinite sum settles on a finite value: 1 divided by 1 minus r.
With r equal to one half, for example, 1 plus a half plus a quarter and so on adds up to exactly 2. When r is 1 or more in size, the series has no finite sum.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| ∑ₖ₌₀^∞ | the sum from k equals 0 to infinity of | add up the terms for every k = 0, 1, 2, … without end |
| r | r | the common ratio, with −1 < r < 1 |
| rᵏ | r to the power of k | r multiplied by itself k times |
Where you meet it
Algebra and calculus courses, compound interest, annuities and probability.
Questions
How do you say Geometric series out loud?
∑ₖ₌₀^∞ rᵏ = 1 / (1 − r) is read: “the sum from k equals 0 to infinity of r to the power of k, equals 1 over the quantity 1 minus r.” That is exactly what Aethel says when it meets the formula in a book.
What does ∑ₖ₌₀^∞ mean in Geometric series?
∑ₖ₌₀^∞ (said “the sum from k equals 0 to infinity of”) is add up the terms for every k = 0, 1, 2, … without end.
What does r mean in Geometric series?
r (said “r”) is the common ratio, with −1 < r < 1.
Can I listen to the equations in my own textbook like this?
Yes. Upload a PDF, EPUB or text file and Aethel reads it aloud with every equation spoken as maths, the way it reads this one, and highlights each sentence as it goes. You can start free.