1 − 2 + 3 − 4 + ⋯W
1 − 2 + 3 − 4 + ⋯

In mathematics, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are the successive positive integers, given alternating signs. Using sigma summation notation the sum of the first m terms of the series can be expressed as

Geometric progressionW
Geometric progression

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2.

Harmonic progression (mathematics)W
Harmonic progression (mathematics)

In mathematics, a harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression.

Lambert seriesW
Lambert series

In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form

Mercator seriesW
Mercator series

In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm:

Divergence of the sum of the reciprocals of the primesW
Divergence of the sum of the reciprocals of the primes

The sum of the reciprocals of all prime numbers diverges; that is:

Sylvester's sequenceW
Sylvester's sequence

In number theory, Sylvester's sequence is an integer sequence in which each term of the sequence is the product of the previous terms, plus one. The first few terms of the sequence are2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443.