Image:Pm1234 Cauchy.png
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[edit] Summary
1 − 2 + 3 − 4 + · · · as the Cauchy product of two copies of 1 − 1 + 1 − 1 + · · ·.
The illustration uses the rectangle metaphor for multiplication. One copy of 1 − 1 + 1 − 1 + · · · is depicted at the top, another at the left. A black length or area represents a positive quantity; a red length or area represents a negative quantity. Multiplying two line segments results in a rectangle whose color is determined by the law for multiplying signed numbers:
- 1 * 1 = 1
- −1 * 1 = −1
- 1 * −1 = −1
- −1 * −1 = 1
The double series resulting from the multiplication of the two single series is expressed as a single series using the Cauchy product rule. Each term of the resulting series is a combination of terms from the double series running from south-west to north-east. The result is identified as 1 − 2 + 3 − 4 + · · ·.
