Key Terms
Sequence
A function whose domain is the positive integers; an ordered list of numbers. Term: a single number in a sequence (a_1,
Explicit formula
Defines a_n directly from n (the position).
Recursive formula
Defines a_n using the term(s) before it.
Special cases
0! = 1 1!
Series
The sum of the terms of a sequence. Partial sum (S_n): the sum of the first n terms of a series.
General form
Sum from k = lower to upper of (formula in k)
To evaluate
1. Substitute each integer value of k from the lower limit to the upper limit into the formula.
Partial sum formula
S_n = a_1(1 - r^n) / (1 - r), where r ≠ 1
How to use it
1. Identify a_1, r, and n.
Finding n when it is not obvious
Use the explicit arithmetic formula: a_n = a_1 + (n-1)d Plug in a_n and d, then solve for n.
Sum of infinite geometric series
S = a_1 / (1 - r), only when |r| < 1
How to determine if the sum exists
1. Check that the series is geometric (constant ratio between consecutive terms).
Explicit arithmetic
A_n = a_1 + (n-1)d Explicit geometric: a_n = a_1 * r^(n-1) Factorial: n! = n * (n-1) * ...
Arithmetic partial sum
S_n = n(a_1 + a_n) / 2 Geometric partial sum: S_n = a_1(1 - r^n) / (1 - r), r ≠ 1
Infinite geometric sum
S = a_1 / (1 - r), only when |r| < 1