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 (1, −2, 3, −4, ...) do not approach 0; therefore 1 − 2 + 3 − 4 + ... 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 1 − 2 + 3 − 4 + ... are:

1,

1 − 2 = 1 - 2,
1 − 2 + 3 = (1 + 3) - 2,
1 − 2 + 3 − 4 = (1 + 3) - (2 + 4),
1 − 2 + 3 − 4 + 5 = (1 + 3 + 5) - (2 + 4),
1 − 2 + 3 − 4 + 5 − 6 = (1 + 3 + 5) - (2 + 4 + 6),

...

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 [x−1, x+1]), so 1 − 2 + 3 − 4 + ... diverges.

Stability and linearity

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