zSnout

Infinity is -1/12

Proof that 1 + 2 + 3 + 4 ... = -1/12.

Did you know that the sum of all whole numbers is equal to -1/12? In this article we find out how it’s possible and prove it ourselves using simple algebra and math!

DISCLAIMER: The math used in this article doesn’t count as real reasoning; it’s just a fun thing to think about. See this video by Mathologer for more information.

The Series

We will define three series that we need to find out: SS, GG, and HH.

S=1+2+3+4+5+...G=11+11+1...H=12+34+5...\begin{aligned} S &= 1 + 2 + 3 + 4 + 5 + ... \\ G &= 1 - 1 + 1 - 1 + 1 - ... \\ H &= 1 - 2 + 3 - 4 + 5 - ... \\ \end{aligned}

Solving for G

Our base step is to solve GG. This is pretty simple:

G=11+11+1...G=1(11+11...)G=1G2G=1G=0.5\begin{aligned} G &= 1 - 1 + 1 - 1 + 1 ... \\ G &= 1 - (1 - 1 + 1 - 1 ...) \\ G &= 1 - G \\ 2G &= 1 \\ G &= 0.5 \\ \end{aligned}

Now we have proved that G=12G = \frac{1}{2}!

Solving For H

Now that we’ve solved for GG, we can use it to solve HH. Here’s the answer for this:

H=12+34+5...2H=12+34+5...+0+12+34...2H =11+11+1...2H=G2H=0.5H=0.25\begin{aligned} H &= 1 - 2 + 3 - 4 + 5 ... \\ 2H &= 1 - 2 + 3 - 4 + 5 ... \\ &+ 0 + 1 - 2 + 3 - 4 ... \\ 2H \text{ } &\overline{= 1 - 1 + 1 - 1 + 1...} \\ 2H &= G \\ 2H &= 0.5 \\ H &= 0.25 \\ \end{aligned}

Now we have proved that H=14H = \frac{1}{4}!

Solving For S

Now that we’ve shows what GG and HH are equal to, it’s time to finally solve for SS.

S=1+2+3+4+5+6...SH=1+2+3+4+5+6...12+34+5+6...SH =0+4+0+8+0+12...SH=4SS0.25=4S1/4=3S1/12=S\begin{aligned} S &= 1 + 2 + 3 + 4 + 5 + 6 ... \\ S - H &= 1 + 2 + 3 + 4 + 5 + 6 ... \\ &- 1 - 2 + 3 - 4 + 5 + 6 ... \\ S - H \text{ } &= 0 + 4 + 0 + 8 + 0 + 12 ... \\ S - H &= 4S \\ S - 0.25 &= 4S \\ -1/4 &= 3S \\ -1/12 &= S \\ \end{aligned}

Now we have proved that S=1/12S = -1/12, and we’re done!

Sources & Resources

I originally found out about this phenomenon at this YouTube video by Numberphile. It’s a great video and you should check it out.