Infinite Series of Real Numbers (Absolute Convergence)

Please see the attached file for the fully formatted problem. Define ak recursively by a1 = 1 and ak = (−1)k  1 + k sin  1 k  −1 ak−1, k > 1. Prove that P 1k =1 ak converges absolutely. Since this problem is an analysis problem, please be sure to be rigorous.

Infinite Series of Real Numbers (Absolute Convergence)

Please see the attached file for the fully formatted problems. Suppose ak  0 and a1/k k ! a as k ! 1. Prove that P 1k =1 akxk converges absolutely for all |x| < 1/a if a 6= 0 and for all x 2 R if a = 0. Since this problem is an analysis problem, please be sure to be rigorous.

Uniform Convergence of an Infinite Series of Functions

Uniform Convergence of an Infinite Series of Functions

Uniform Convergence of an Infinite Series of Functions

Uniform Convergence of an Infinite Series of Functions

Uniform Convergence of the Geometric Series

Uniform Convergence of the Geometric Series