How exactly might one go about showing that(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\left| \frac{1 - e^{-iy}}{-iy} \right|

[/tex]

is bounded by 1 for [itex]y\in \mathbb R[/itex]? I thought this would be easy to show using the series expansion of [itex]e^{-iy}[/itex] in some way:

[tex]

\left| \frac{1 - e^{-iy}}{-iy} \right| = \left| 1 - \frac{iy}{2} - \frac{y^2}{6} + \frac{iy^3}{24} + \ldots \right| = \lim_{N\to \infty} \left| \sum_{n=0}^N \frac{(-iy)^n}{(n+1)!} \right| = \lim_{N\to \infty} g(y,N).

[/tex]

But this doesn't seem to be the right way to do it. For example, if you look at [itex]g(8,N)[/itex] for the first several values of [itex]N\in \mathbb N[/itex], it eventually settles down to something less than one, but it takes it awhile! Indeed, one needs to take [itex]N=14[/itex] to make [itex]g(8,N) < 1[/itex]. So I'm not sure a simple "inequalities are preserved under the limit" argument will work. And the series isn't alternating, so the alternating series estimation theorem won't work, either. Any ideas?

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# Why is this expression bounded by 1?

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