Series of Constants

Arithmetic Series

a + (a + d) + (a + 2d) + ... + {a + (n - 1)d} = (1/2).n.{2a + (n - 1)d} = (1/2).n.(a+ l)
where l = a + (n - 1)d is the last term.

Some special cases are
1 + 2 + 3 + ... + n = (1/2).n.(n + 1)
1 + 3 + 5 + ... + (2n - 1) = n²

Geometric Series

a + ar + ar² + ar³ + ... + arⁿ = a(1 - rⁿ)/(1 - r) = (a - rl)/(1 - r)
where l = ar^{n - 1} is the last term and r ≠ 1.

If - 1 < r < 1, then
a + ar + ar² + ar³ + ... + arⁿ = a/(1 - r)

Arithmetic - Geometric Series

a + (a + d)r + (a + 2d)r² + ... + {a + (n - 1)d}r^{n - 1} =

 
where r ≠ 1.

If -1 < r < 1, then
a + (a + d)r + (a + 2d)r² + ... =

Sums of Powers of Positive Integers

where the series terminates at n² or n according as p is odd or even, and B_k are the Bernoulli numbers.

Some special cases are

If S_k = 1^k + 2^k + 3^k + ... + n^k where k and n are positive integers, then

Series Invoving Reciprocals of Powers of Positive Integers

Miscellaneous Series

The Euler - Maclaurin Summation Formula

The Poisson Sommation Formula

 

Miscellaneous Series