Calculus Answers

Suppose lim n→∞ (sn −1)/(sn +1)=0. Prove that limn→∞sn = 1

Prove that if a sequence {an}∞n=1 satisfies Cauchy’s criterion, then it is bounded

Let {an}∞n=1 and {bn}∞n=1 be sequences both of which diverge to −∞. Then {an +bn}∞n=1 diverge to −∞ and {an ·bn}∞n=1 diverge to +∞

find "y'" for each:

1. "y= x\u00b3 sin x"

"y= x\u00b2 + 2x cos x"2. "y= x\u00b2 + 2x" "cos x"

3."y = x\/( sec x + 1)"

"y= x\u00b2 + 2x cos x"

Let {an}∞n=1 and {bn}∞n=1 be sequences both of which diverge to +∞. Then both of {an +bn}∞n=1 and {an · bn}∞n=1 diverge to +∞

Prove that a non-decreasing (resp. non-increasing) sequence which is not bounded above (resp. bounded below) diverges to +∞ (resp. to −∞)

Prove that a sequence that diverges to +∞ (resp. −∞) is divergent according to the previous definition

find "y'" for each:

1. "y= sin x - cos x"

2. "y= tan x \/ (x+1)"

3."y=sin (\u03c0\/4)"

Let {an}∞n=1 be a convergent sequence with limit L such that an ≤ B forevery n≥N for some N ∈ N and some B ∈ R. Prove that L ≤ B

Let {an}∞n=1 be a non-decreasing (resp. non-increasing) sequence which converges to a. Then prove that an ≤ a (resp. a ≤ an) for every n ∈ N