Unusual ways of summing well-known series -- for example, this unusual summation of the geometric series The typical way of summing $S_g(n,x) = 1+x+x^2+\cdots+x^n$ by multiplying by $(1-x)$ is well known.
The arithmetico-geometric series $S_{ag}(n,x) = 1+2x+3x^2+4x^3+\cdots+(n+1)x^n$ can be summed in one of two ways: 1) apply $(1-x)$ twice, or 2) notice that the $n$th term is the derivative of $x^n$, and thus that  $\frac{d}{dx}S_g(n,x) = S_{ag}(n,x)$. Let's call the first method the "multiplication" method and the second method the "differentiation" method.
$S_g$ can be summed using the multiplication method, and $S_{ag}$ can be summed using both the multiplication and the differentiation methods (assuming that you know the sum of $S_g$). The natural question is whether or not $S_g$ can also be summed using the differentiation method or something like it. This led me to the following summation of the geometric series:
Multiply by $e^{yx}$ (which is never zero) on both sides to get
$$
S_g(n,x)e^{yx} = e^{yx} + xe^{yx}+x^2e^{yx}+\cdots + x^ne^{yx}. 
$$
Noting that $xe^{yx} = \frac{\partial}{\partial y}e^{yx}$, it follows that
\begin{align}
e^{yx} + xe^{yx}+x^2e^{yx}+\cdots + x^ne^{yx} &= e^{yx} +  \frac{\partial}{\partial y}e^{yx} + x\frac{\partial}{\partial y}e^{yx}+\cdots+x^{n-1}\frac{\partial}{\partial y}e^{yx} \\
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&=e^{yx}+ \frac{\partial}{\partial y}S_g(n-1,x)e^{yx} \\
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&=e^{yx}+ \frac{\partial}{\partial y}[S_g(n,x)e^{yx}-x^ne^{yx}] \\
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&=e^{yx}+\frac{\partial}{\partial y}S_g(n,x)e^{yx}-x^{n+1}e^{yx}, 
\end{align}
and thus,
$$
S_g(n,x)e^{yx}-\frac{\partial}{\partial y}S_g(n,x)e^{yx} = (1-x^{n+1})e^{yx}.
$$
Finally, since $\frac{\partial}{\partial y}S_g(n,x)e^{yx} = xS_g(n,x)e^{yx}$, it follows that
$$
(1-x)S_g(n,x)e^{yx} = (1-x^{n+1})e^{yx},
$$
and hence, for all $x\neq 1$, it follows that
$$
S_g(n,x) = \frac{1-x^{n+1}}{1-x}.
$$
This method is clearly unusual in that, although it achieves the correct result, it uses more machinery (calculus, exponentials), and is a bit more complicated than the typical way. Nonetheless, I also think there's something fascinating about seeing all the different ways a series can be summed.
My general question is what other "unusual" ways of summing well-known series do people know? My specific question is whether or not this particular way of summing the geometric series is known?
My guess for the second question is 'yes', because the tools are still pretty basic and the manipulations not that complicated, but I've only ever seen the typical way.
 A: When I was in high school a friend showed me a way to sum the infinite arithmetico-geometric series, made a big impression on me at the time: $$\matrix{1&+&x&+&x^2&+&x^3&+&x^4&+&\cdots\cr&&x&+&x^2&+&x^3&+&x^4&+&\cdots\cr&&&&x^2&+&x^3&+&x^4&+&\cdots\cr&&&&&&\vdots&&\vdots&&\ddots}$$ If you sum the columns, you get $1+2x+3x^2+4x^3+\cdots$. If you sum the rows, you get $${1\over1-x}+{x\over1-x}+{x^2\over1-x}+\cdots$$ a geometric series with sum ${1/(1-x)\over1-x}=(1-x)^{-2}$, and we're done. OK, we should insist on $|x|<1$ to guarantee convergence and justify the manipulations, and we should realize that all that's going on here is interchange of summations, but it's still pretty neat.
A: Another way to get to series is through integration. The most known example is probably
$$
\sum_{n\geq 1} \dfrac{1}{n^2} = \int_0^1\int_0^1 \dfrac{1}{1 - xy}\,dx\,dy
$$
which often serves as exercise for multiple integration and change of variables. Another example has recently been presented in this video by Michael Penn on his very nice channel. I hope this is what you had in mind!
