1. Math
  2. Advanced Math
  3. 2 verify that the following definitions are valid the limits...

Question: 2 verify that the following definitions are valid the limits...

Question details

2. Verify that the following definitions are valid (the limits exists for each x) sin(z) (-1)(k-1)/2k! Σ (-1)k/2 k! and cos(x) k-1, k odd k-0, keven and that dx A Road Map to Glory: You may assume throughout this exercise that, for any aER converges to e for all a E IR k! (a) Show that, for a 0 and x E-a, a (-1)(k-1)/2-l〈 ak (b) Show that Xk-lk odd coverges by showing that it is strictly less than ea (c) What theorem guarantees that the series representation weve defined to be sin(x) converges, for all x E-a, a and for any arbitrary a > 0 (which is to say for all xER) d) Show that the series k! k-1, k odd is equal to the series representation for cos(x) and that this series converges uniformly, using similar reasoning as in problems (a) - (c) (e) Which theorem then confirms that (sin x) = cosx?

Solution by an expert tutor
Blurred Solution
This question has been solved
Subscribe to see this solution