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There is a related question here: Statement about existence of a polynomial - true or false?

The question is from Advanced Problems in Core Mathematics by Siklos and is stated as follows (determine if true or false, then prove):

There exists a polynomial $P$ such that $|P(x)-\cos x|\le10^{-6}$ for all real $x$.

Now, I've read the solution and am still a bit unhappy. What I don't really get is that the answer relies on the argument that a polynomial can be made arbitrarily large by letting $x$ become very large. Given we can approximate $cos(x)$ as $P(x)$ with a Maclaurin expansion, can't we just choose an infinite amount of terms in our expansion and thus approximate $cos(x)$ to the desired degree of accuracy across the whole real line?

Obviously I'm aware that the answer is false (and have read the solution several times) so I'm interested in a nice explanation that clears up my confusion.

Thanks,

Mark

P.S. Book here: http://cuhs.co.uk/wp-content/uploads/2016/03/advpcm.pdf

P.P.S. It's hinted in the related question that a polynomial has finite degree but our Maclaurin expansion has infinite terms.

P.P.P.S Bonus question on why the $M^{1/n}$ is needed in the proof given in the book, why not just $M$?

User123456789
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It is implied that the polynomial $P$ has finite degree – whereas the Maclaurin expansion of $\cos x$, though true for all $x$, has infinite degree. As such, the argument of unboundedness applies to $P$ and the statement is false.

Parcly Taxel
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  • Thanks for the (very fast!) reply. Perhaps I'm being an idiot but how is it implied that the polynomial has finite degree? In the Comments Siklos immediately starts talking about the expansion (albeit finite) of $cos(x)$. – User123456789 Sep 26 '17 at 13:41
  • @maw501 A polynomial, when stretched out to infinite degree, is almost never called a polynomial any longer. It becomes a power series. It is a matter of nomenclature. – Parcly Taxel Sep 26 '17 at 13:43
  • @maw501 Also, do you know about accepting answers already? – Parcly Taxel Sep 26 '17 at 13:46
  • Thank you for the clarification (yes, just accepted! Though I was busy adding a slight extension to it). – User123456789 Sep 26 '17 at 13:50