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I'd like to put together a compilation of visually geometric proofs of series summations. I have three famous 2D examples to clarify what I mean below, but other "visually geometric" proofs of an infinite sum are welcome. If you can add to this list with a picture, a link to the internet somewhere, or some other reference, that helps. I can't pick the correct answer to a question like this, so I'll make this community wiki.


$$\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+1}\right)=1$$

enter image description here using the observation $\left(\frac{1}{n}-\frac{1}{n+1}\right)=\int_0^1(x^{n-1}-x^n)\,dx$


$$\sum_{n=0}^{\infty}\left(\frac{1}{2n+1}-\frac{1}{2n}\right)=\ln(2)$$

enter image description here using that $\ln(2)=\int_0^1\frac{1}{x+1}\,dx$. Each point in this image has $x$-coordinate a fraction with a power of $2$ for its denominator, and $y$-coordinate determined by the curve $y=\frac{1}{x+1}$.


$$\sum_{n=1}^\infty\frac{n}{2^{n+1}}=1$$

enter image description here where the large square is $1\times1$ and for each $n$ we have $n$ rectangles of area $\frac{1}{2^{n+1}}$.

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