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:This is a good time to remember that we're here to write an encyclopedia, [[WP:NOTTEXTBOOK|not a textbook]]. It is often better for our purposes to summarize a proof concisely in prose, rather than to step through it equation-by-equation. Accordingly, {{u|Deacon Vorbis}}'s suggestion here is reasonable. [[User:XOR'easter|XOR'easter]] ([[User talk:XOR'easter|talk]]) 18:23, 22 December 2019 (UTC)
:: I agree to delete most details, but not the key details. Otherwise, the reader may not be able to do the proof himself if he wants to. Here is what I would write:
::In 1741, Euler published a second proof that did not rely on infinite products. In it, he computes the integral <math> \int ^{1}_{0}\frac{\arcsin( x)}{\sqrt{1-x^{2}}} \ dx </math> by two methods: first directly, and then by expanding the arcsine as its [[Taylor series]] and integrating term-by-term. He then computes <math>\int ^{1}_{0}\frac{x^{2n+1}}{\sqrt{1-x^{2}}} \ dx</math> using an integration by parts. Equating the two gives the value of
::<math>\sum ^{+\infty }_{n=0}\frac{1}{( 2n+1)^{2}}</math>. He finally separates odd and even numbers in <math>\sum ^{+\infty }_{n=1}\frac{1}{n^{2}}</math> to complete the proof.
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