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Euler's infinite series

Just wanted to know whether Euler was correct in assuming the laws of a finite polynomial to hold true even for an infinite polynomial when he working was on the infinite sum

1+1/4+1/9+1/16 .........

Or in other words is Euler justified in expanding an infinite polynomial into product of infinite roots as he did with the sin x/x case

And is he justified in equating the co-efficientls of infinite polynomials.as far as I have seen funny things happen when we apply rules of finite series to infinite series.

So is the proof that Euler gave us rigorous or is there a more rigorous proof.

Note by Pranav Chakravarthy
4 years ago

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There's a more rigorous proof involving Fourier series. Jimmi Simpson · 4 years ago

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