5 The Ramanujan Summation

In a letter to Hardy in 1913, Ramanujan wrote: "Dear Sir, I am very much gratified on perusing your letter of the 8th February and am now sending you a few more of my results..." Among them was the assertion that the sum of all natural numbers equals -1/12.

Define three series. First, the Grandi series:

S1=1−1+1−1+⋯=12S_1 = 1 - 1 + 1 - 1 + \cdots = \frac{1}{2}

Next, consider:

S2=1−2+3−4+5−⋯S_2 = 1 - 2 + 3 - 4 + 5 - \cdots

Shift and add S_2 to itself:

2S2=1−1+1−1+⋯=S1=122S_2 = 1 - 1 + 1 - 1 + \cdots = S_1 = \frac{1}{2}
S2=14S_2 = \frac{1}{4}

Now let S = 1 + 2 + 3 + 4 + ... and compute S - S_2:

S−S2=0+4+0+8+0+12+⋯=4SS - S_2 = 0 + 4 + 0 + 8 + 0 + 12 + \cdots = 4S

Therefore -3S = S_2 = 1/4, and so:

∴
  1+2+3+4+⋯=−112  \boxed{\;1 + 2 + 3 + 4 + \cdots = -\frac{1}{12}\;}

This result, properly understood via analytic continuation of the Riemann zeta function, appears in string theory and the calculation of the Casimir effect.