1 − 2 + 3 − 4 + ⋯

In mathematics, 1 − 2 + 3 − 4 + ··· is an infinite series. Using sigma summation notation the sum of the first m terms of the series can be expressed as$$\sum _{n=1}^{m}n(-1{)}^{n-1}.$$

Divergence

The series' terms Template:Nowrap do not approach 0; therefore Template:Nowrap diverges by the term test. Divergence can also be shown directly from the definition: an infinite series converges if and only if the sequence of partial sums converges to limit, in which case that limit is the value of the infinite series. The partial sums of Template:Nowrap are: Template:Block indent The sequence of partial sums shows that the series does not converge to a particular number: for any proposed limit x, there exists a point beyond which the subsequent partial sums are all outside the interval Template:Nowrap), so Template:Nowrap diverges.

Stability and linearity

Since the terms Template:Nowrap follow a simple pattern, the series Template:Nowrap can be manipulated by shifting and term-by-term addition to yield a numerical value. If it can make sense to write Template:Nowrap for some ordinary number s, the following manipulations argue for Template:Nowrap$$\begin{array}{ccccccc}4s& =& & (1-2+3-4+\cdots )& +(1-2+3-4+\cdots )& +(1-2+3-4+\cdots )& +(1-2+3-4+\cdots )\\ & =& & (1-2+3-4+\cdots )& +1+(-2+3-4+5+\cdots )& +1+(-2+3-4+5+\cdots )& +(1-2)+(3-4+5-6\cdots )\\ & =& & (1-2+3-4+\cdots )& +1+(-2+3-4+5+\cdots )& +1+(-2+3-4+5+\cdots )& -1+(3-4+5-6\cdots )\\ & =& 1+& (1-2+3-4+\cdots )& +(-2+3-4+5+\cdots )& +(-2+3-4+5+\cdots )& +(3-4+5-6\cdots )\\ & =& 1+[& (1-2-2+3)& +(-2+3+3-4)& +(3-4-4+5)& +(-4+5+5-6)+\cdots ]\\ & =& 1+[& 0+0+0+0+\cdots ]\\ 4s& =& 1\end{array}$$So $s=\frac{1}{4}$.