Prove that there cannot be a sequence of polynomial $p_n$ converging uniformly to $\cos$ or $\sin$ on $\Bbb R$.
Doesn't the Taylor series completely contradict this question?
2026-04-01 11:15:41.1775042141
Is this question wrong? Sequence of polynomial approximates $\sin$
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The Taylor series of a bounded function that converges on all of $\mathbb{R}$ never converges uniformly to that function because the difference between every partial sum of the series and some point in its limit is as large as you want to make it. Polynomials increase in absolute value without bound as we stray far enough from 0.