Find the least upper bound if it exists and the greatest

Question

(1) TRUE/FALSE: If lim_n→∞a_n = 0, then _n=0∑^∞a_n converges.              

(a) True

(b) False

(2) TRUE/FALSE: Since sequence {cos(nπ)}_n=1^∞ is bounded, it will converge.

(a) True

(b) False

(3) If a telescoping series has Mth partial sum s_M = 4 +(1/M-4), what is the sum of the series?

(a) 4      

(b) 0

(c) The series diverges.

BOUNDED SET and AEQUENCES

(4) Find the least Upper bound (if it exists) and the greatest lower bound (if it exists) for the given set.

i) {x:|x-2|≤5}

ii) {1, 1/2, 1/3, 1/4,· · ·}

(5) The first several terms of a sequence {a_n} are given. Assume the pattern continues as indicated and find an explicit formula for a_n.

{-1, 2/3, -1/3, 4/27, -5/81,· · ·}

(6) Determine the boundedness and monotonicity of the sequence with a_n as indicated.

(i) a_n = (n+(-1)ⁿ/2n)

(ii) an = 2/n

CONVERGENCE OF SEQUENCES

(7) State whether the sequence converges as n →∞ and, if it does, find the limit.

(i) a_n = 2 Inn - ln(n²+n)  

(ii) a_n = (1 - 1/n)ⁿ

NUMERICAL INTEGRATION

(8) Estimate

₀∫³ (1/1+x³)dx

by: (a) the kit endpoint rule n = 6.;

(b) the right endpoint rule, n =6;

(c) the midpoint rule, n = 6. What value does your calculator give?

(9) Find the smallest values of n which will guarantee a theoretical error less than ∈ if the integral is estimated by:

(a) the trapezoidal rule;

(b) Simpson's rule.

₁∫⁴√x dx, ∈ = 0.001.