Partial sum
The sum of the first n terms of a series. The infinite series equals the limit of its partial sums, if that limit exists.
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Analogy
A running total on a till receipt: the final bill is what the running total settles on.
Example
S_1 = 1. S_2 = 1.5.S_3 = 1.75. S_4 = 1.875.Totals creep towards 2.So the series sums to 2.
In practice
Defines what it means for an infinite sum to have a value.
Practice
Flashcards
- What is a partial sum?The sum of the first n terms of a series.
Interview questions
- Find the sum of 3 + 1.5 + 0.75 + ... and explain why it is valid.
- Geometric with a = 3, r = 0.5
- |r| < 1 so the series converges
- Sum = a/(1 - r) = 3/0.5 = 6
- Partial sums approach 6
Questions people ask
How can infinitely many positive numbers add to a finite total?
If the terms shrink quickly enough, the running total stays below a ceiling. Adding 1, then 1/2, then 1/4 never passes 2. Each new term fills half of the remaining gap, so the total gets as close to 2 as you like.
What does it mean if the partial sums diverge?
It means the running total never settles on one number. It may grow without bound, like 1 + 2 + 3 and so on, or oscillate, like 1 - 1 + 1 - 1. In either case the series has no sum.