1 − 1 + 2 − 6 + 24 − 120 + …
In mathematics, is a divergent series, first considered by Euler, that sums the factorials of the natural numbers with alternating signs. Despite being divergent, it can be assigned a value of approxi
History of Grandi's series
Guido Grandi (1671–1742) reportedly provided a simplistic account of the series in 1703. He noticed that inserting parentheses into 1 − 1 + 1 − 1 + · · · produced varying results: either or Grandi's e
Harmonic series (mathematics)
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: The first terms of the series sum to approximately , where is the natural logarithm and is the
Divergent geometric series
In mathematics, an infinite geometric series of the form is divergent if and only if | r | ≥ 1. Methods for summation of divergent series are sometimes useful, and usually evaluate divergent geometric
Divergent series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converge
Summation of Grandi's series
The formal manipulations that lead to 1 − 1 + 1 − 1 + · · · being assigned a value of 1⁄2 include:
* Adding or subtracting two series term-by-term,
* Multiplying through by a scalar term-by-term,
*
1 + 2 + 3 + 4 + ⋯
The infinite series whose terms are the natural numbers 1 + 2 + 3 + 4 + ⋯ is a divergent series. The nth partial sum of the series is the triangular number which increases without bound as n goes to i
1 + 2 + 4 + 8 + ⋯
In mathematics, 1 + 2 + 4 + 8 + ⋯ is the infinite series whose terms are the successive powers of two. As a geometric series, it is characterized by its first term, 1, and its common ratio, 2. As a se
1 − 2 + 4 − 8 + ⋯
In mathematics, 1 − 2 + 4 − 8 + ⋯ is the infinite series whose terms are the successive powers of two with alternating signs. As a geometric series, it is characterized by its first term, 1, and its c
1 + 1 + 1 + 1 + ⋯
In mathematics, 1 + 1 + 1 + 1 + ⋯, also written , , or simply , is a divergent series, meaning that its sequence of partial sums does not converge to a limit in the real numbers. The sequence 1n can b
Grandi's series
In mathematics, the infinite series 1 − 1 + 1 − 1 + ⋯, also written is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatme
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