Show your work and give clear explanations and proofs in all problems. If you use a theorem, state the theorem 3. (34 pts) Assume that (an) is a sequence in R and an > 0 for all n in N. Prove that if an converges, then n+1 also converges. nel
Please prove this, thanks! 2. Let {xn n21 be a sequence in R such that all n > 0. If ( lim supra) . (lim supー) = 1 Tn (here we already assume both factors are finite), prove that converges.
2. Show that for > 0, we have x4 + IV
Prove that is an integer for all n > 0.
HW: Show that the series __, an n=0 converges whenever ſal < 1, and diverges whenever al > 0.
(10pts) 3. Use direct proof to show that if x and y are positive real numbers, then (x+y)" > " + y".
Let a1 = 3 and an+1 = a + 5 2an for all n > 1 Prove that (an)nen converges and find limn7oo an:
For all n E N prove that 0 <e- > < 2 k!“ (n + 1)! k=0 Hint: Think about Taylor approximations of the function e".
bn converges 18. Let (an)n=1 and (bn)n=1 be sequences in R. Show that if and lan – an+1 < oo, then anbr converges.
Let n ez, n > 0; let do, d1,..., dn, Co,..., En be integers in the range {0, 1, 2, 3,4}. Prove: If 5*dx = 5* ex k=0 k=0 then ek = =dfor k = 0,1,...,n.