undergraduate level problem 1. For any monotonically decreasing sequence {a_k }_(k=1) ^m , show that Σ a_m converges if and only if Σ 2^m a_2^m converge. additionally prove that Σ (1/m^s ) converges and only if s>1. 2.Provide a counterexample for which lim_(n→∞) ∫_E f_n (x)dx= ∫_E lim_(n→∞) f_n (x)dx does not hold. 3. show that Weiestrass function f(x)=Σ_(k=0) ^∞ a^k cos(b^k πx) , 1+((3π)/2)<ab is continuous function at x∈[−M,M] 4. prove f(x) is not differentiable x∈R 5. 1_Q = { (( 1 , x∈Q)),((0 , x∈R/Q)) :} is Riemann integrable ?? Source:Introduction to Analysis SNU (Seoul National University of Kor)major text books did you try it?? :) and these are the legandary math Bibles of SNU