Question

2. Prove that the infinite series Ex=1(-1)k diverges. (Hint: Compute the first few terms of the sequence of partial sums and determine a formula for the nth partial sum, Sn. Using this, give a formal proof, starting with the definition for divergence of this series. (Additional reference: Workshop Week #7)

Answer 1

E (-1) K=1 partial sum of the above series 5=-1 S2=-1+1=0 =-1+1-1=-1 sn = { -1 ; if his odd if n in even The -1,0,-1,0, s Sequence of partial sum clearly Sn S to Then Sn converges does not converges (hence diverges) If a number every subsequence of Sn converges to s even subsequence odd subsequence which shows Szo and I but oz-1 Convergest S Szn = (0) in particular S21+1=-1 S Hence Sn is not convergent ( divergent)