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## Homework Statement

Let g_n : R -> R be given by g

_{n}(x) := cos

^{2n}(x), does g

_{n}

^{'}converge uniformly?

## Homework Equations

The derivative is as follows; -2nsin(x)cos

^{2n-1}, which I have found converges pointwise to the 0 function.

Formal definition of Uniform Convergence;

For all e>0, there exits N(e) in the natural numbers, such that for all n >= N, for all x in R,

imples Mod( f

_{n}(x) - f(x)) < e

where f is the limit of the sequence of functions f

_{n}

## The Attempt at a Solution

I have been struggling at seeing how to prove functions are uniformly convergent in general, but have been told that this does not converge uniformly, however I really cannot see how this is true, as

Supremum norm(g

_{n}

^{'}- 0) ->0 as n-> Infinity,

which is an equivalent definition of uniform convergence, because the term cos

^{2n-1}is dominant over 2n.

Thanks for any help :)