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dodonaomikiki

Euler's original derivation of the value
π2
/
6
essentially extended observations about finite polynomials and assumed that these same properties hold true for infinite series.

Of course, Euler's original reasoning requires justification (100 years later, Karl Weierstrass proved that Euler's representation of the sine function as an infinite product is valid, by the Weierstrass factorization theorem), but even without justification, by simply obtaining the correct value, he was able to verify it numerically against partial sums of the series. The agreement he observed gave him sufficient confidence to announce his result to the mathematical community.

To follow Euler's argument, recall the Taylor series expansion of the sine function

dodonaomikiki

帕塞瓦尔恒等式 编辑

在数学分析中，以Marc-Antoine Parseval命名的帕塞瓦尔恒等式是一个有关函数的傅里叶级数的可加性的基础结论。表示可积函数与其傅里叶系数之间关系的恒等式。从几何观点来看，这就是内积空间上的毕达哥拉斯定理。
它由帕塞瓦尔(Parseval，C.M.-A.)于1805年提出但未证明。对于黎曼可积函数情形是李亚普诺夫(Ляпунов，А.М.)于1896年证明的。1906年，勒贝格(Lebesgue，H.L.)对于勒贝格平方可积函数给出证明




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en.m.wikipedia.org/wiki/Parseval%27s_identity