Concept
1 − 2 + 3 − 4 + ⋯
Related concepts (7)
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 series of real numbers it diverges to infinity, so the sum of this series is infinity. However, it can be manipulated to yield a number of mathematically interesting results. For example, many summation methods are used in mathematics to assign numerical values even to a divergent series.
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 infinity. Because the sequence of partial sums fails to converge to a finite limit, the series does not have a sum. Although the series seems at first sight not to have any meaningful value at all, it can be manipulated to yield a number of mathematically interesting results.
1 + 1 + 1 + 1 + ⋯
In mathematics, 1 + 1 + 1 + 1 + ⋯, also written \sum_{n=1}^{\infin} n^0, , 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 be thought of as a geometric series with the common ratio 1. Unlike other geometric series with rational ratio (except −1), it converges in neither the real numbers nor in the p-adic numbers for some p. In the context of the extended real number line since its sequence of partial sums increases monotonically without bound.
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 treatment of the series in 1703. It is a divergent series, meaning that it does not have a sum. However, it can be manipulated to yield a number of mathematically interesting results. For example, many summation methods are used in mathematics to assign numerical values even to a divergent series.
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 converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme.
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 common ratio, −2. As a series of real numbers it diverges, so in the usual sense it has no sum. In a much broader sense, the series is associated with another value besides ∞, namely 1/3, which is the limit of the series using the 2-adic metric. Gottfried Leibniz considered the divergent alternating series 1 − 2 + 4 − 8 + 16 − ⋯ as early as 1673.
Geometric series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series is geometric, because each successive term can be obtained by multiplying the previous term by . In general, a geometric series is written as , where is the coefficient of each term and is the common ratio between adjacent terms.