Uniform Convergence?

Question

I need to show that fn(x)=cos(x/n) converges uniformly to f(x)=1 for x in [0, pi]. I need help finding an n from the inequality
|cos(x/n) - 1| < eps

Thanks...

Answer 1

Use the Taylor series expansion for cos(x/n). For n sufficiently large, this is an alternating series whose terms decrease in absolute value (use x=pi to guarantee that this is so for all x in [0,pi]). For such a series, the error is less than the absolute value of the first term left off. In this case, use only the first term of the series; then the error is less than pi^2/(2n^2)

Answer 2

Look at the Taylor series for cos (t), then you should be able to show that 1 - cos(t) is smaller than [t^2]/2 and things should be easy to find an appropriate n in terms of epsilon.