-1/12 is a large friggin’ number

Suppose $$S_n\equiv\sum^n_{i=1}{i}=1+2+\cdots+\left(n-1\right)+n$$ You may have figured out that you can rearrange the sum as $$S_n=n+\left(n-1\right)+\cdots+2+1$$ This is interesting because the first terms in each arrangement sum to $n+1$, the second terms also sum to $n-1+2=n+1$. In fact, this is true of all $n$ terms so we find that $$S_n+S_n=n\left(n+1\right)$$ Fine. But what about $S_\infty$? The sum of all natural numbers, it is obviously bigger than any natural number. Can we more precisely describe $S_\infty$? Perhaps.

Let’s start with something easier. Suppose $$T_n\equiv\sum^n_{i=0}{\left(-1\right)^i}=1-1+1-1+\cdots+\left(-1\right)^n$$ Obviously, $T_n=1$ if $n$ is odd, and $T_n=0$ if $n$ is even. What is $T_\infty$?

Consider the polynomial $$t\left(x\right)\equiv\sum^\infty_{i=0}{\left(-1\right)^ix^i}=1-x+x^2-x^3+\cdots$$ so that $t\left(1\right)=1-1+1-1+\cdots=T_\infty$. However, $$t\left(x\right)=1-x\left(1-x+x^2-x^3+\cdots\right)=1-xt\left(x\right)$$ which implies $t\left(x\right)=1/\left(1+x\right)$. Consequently, $$T_\infty=t\left(1\right)=\frac{1}{2}$$ Now what? Let us move on to something trickier. Consider $$U_n\equiv\sum^n_{i=0}{\left(-1\right)^ii}=0-1+2-3+\cdots+\left(-1\right)^nn$$ Following our previous strategy, we define $$u_0\left(x\right)\equiv\sum^\infty_{i=0}{\left(-1\right)^iix^i}=0-1x+2x^2-3x^3+\cdots$$ so that $u_0\left(1\right)=0-1+2-3+\cdots=U_\infty$. Let us also define $$u_1\left(x\right)\equiv\sum^\infty_{i=1}{\left(-1\right)^{i-1}\left(i-1\right)x^i}=0x-1x^2+2x^3-3x^4+\cdots$$ so that $u_1\left(1\right)=0-1+2-3+\cdots=U_\infty$. This means that $2U_\infty=u_0\left(1\right)+u_0\left(1\right)$. But $$u_0\left(x\right)+u_1\left(x\right)=0-1x+1x^2-1x^3+1x^4\cdots=t\left(x\right)-1$$ so $u_0\left(1\right)+u_0\left(1\right)=t\left(1\right)-1=-1/2$ and therefore $$U_\infty=\frac{1}{2}\left[u_0\left(1\right)+u_0\left(1\right)\right]=-\frac{1}{4}$$ Okay. That’s kind of weird, but where does that get us?

Once more, we define $$s\left(x\right)\equiv\sum^\infty_{i=0}{ix^i}=0+1x+2x^2+3x^3+\cdots$$ so that $s\left(1\right)=1+2+3+\cdots=S_\infty$. Noting $$s\left(x\right)+u_0\left(x\right)=0+0x+4x^2+0x^3+8x^4+\cdots=4s\left(x^2\right)$$ and specifically that $s\left(1\right)+u_0\left(1\right)=4s\left(1\right)$ we find $$S_\infty=s\left(1\right)=\frac{1}{3}u_0\left(1\right)=\frac{1}{3}U_\infty=-\frac{1}{12}$$ Did we not already establish that $S_\infty$ is larger than any natural number? Apparently, $-1/12$ is a very, very large number.