So, I have a solution, but something is seriously off. If anyone gets a chance please correct any logical errors you may come across.
Ok, so I estimated cos u > .995 for 0<= t <= .1.
Then, I set up the difference form of the triangle inequality:
|1/2*cos (t) + 1/4*cos(2t)+..+1/2^k*cos(kt)|
>= |1/2||cos(t)| - |1/4||cos(2t)|-..-|1/2^k||cos(kt)|.
>=1/2*.995 - 1/4*.995-...-1/2^k*.995 (for 0 <= t <= .1)
>=.7 ????--> (this is supposed to be my conclusion, but clearly this step is not true, or am I not seeing something).