Solving a heat equation with central symmetry i got the following result:
The problem is to find a sphere center temperature vs. time given that the surface of the sphere is kept constant at $$T_1$$ and at initial moment t=0 the temperature inside the sphere is distributed uniformly with value $$T_0$$
A solution is following:
$$T(t)=T_1+2 (T_0-T_1)\sum_{k=1}^\infty (-1)^{k+1} e^{-t(\frac{k\pi a}{r})^2}$$ Here r is the radius of the sphere and a=const is a coefficient of thermal conductivity.
Now the solution must satisfy the initial condition $$T(0)=T_0$$:
$$T(0)=T_1+2 (T_0-T_1)\sum_{k=1}^\infty (-1)^{k+1} $$
This is satisfied only if the following holds:
1-1+1-1+1-1+ ... =1/2
So, we found this divergent sum through physical arguments? Is this appropriate for the pure mathematician?
Lets go a step further and differentiate the solution by t:
$$\frac {d T(t)}{dt}=-2 (T_0-T_1)(\frac{\pi a}{r})^2\sum_{k=1}^\infty (-1)^{k+1} k^2 e^{-t(\frac{k\pi a}{r})^2}$$
Now, at initial moment t=0 we have:
$$\frac {d T(0)}{dt}=-2 (T_0-T_1)(\frac{\pi a}{r})^2\sum_{k=1}^\infty (-1)^{k+1} k^2 $$
Obviously $$\frac {d T(0)}{dt}=0$$ because the maximum speed in the universe is limited.
Thus the following must be valid:
1^2 - 2^2 + 3^2 - 4^2 + ... =0
Here we can say what has been said previously, i think. But at the moment i'm not sure about the following divergent sum:
1 - 2 + 3 - 4 + ...