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.

Easy Math Editor

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Comments

There's a more rigorous proof involving Fourier series.

Jimmi Simpson - 8 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$