A Pure And An Applied Mathematician

A pure and an applied mathematician are asked to calculate 2 * 2.
The applied mathematician's solution: We have
2 * 2 = 2 *1/(1-1/2).
The second factor on the right hand side has a geometric series expansion
1/(1-1/2) = 1 + 1/2 +1/4 + 1/8 + ....
Cutting off the series after the second term yields the approximate solution
2 * 2 = 2 *(1 +1/2) = 3.
The pure mathematician's solution: We have
2 * 2 = (-2) *1/(1-3/2).
The second factor on the right hand side has a geometric series expansion
1/(1-3/2) = 1 + 3/2 +9/4 + 27/8 + ...,
which diverges. Hence, the solution to 2 * 2 does not exist.