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Calculus Analysis Mathematics Question


1. We dene a sequence (an), n in N recursively by

a_1 = -2 and a_(n+1) = -sqrt( 4 -a_n)

Note that (an) n in N is well-defined and we have an < 0 for all n in N

A. Show that (an), n in N is decreasing and bounded below.

Hint: Show that -3 < an+1 < an, for all n in N by induction.

B. Show that (an) n in N converges. Moreover, compute lim n->oo. You can use part 1, even if you did not solve it.

2.Let (an) n in N be a bounded sequence. We set L = lim sup( an); l = lim inf (an):

A. Show that L; l belong to R.

B. Show that for all ϵ > 0, there exists n0 belong to N such that for all n > n0, we have l – ϵ < an < L + ϵ: You can use part 1, even if you did not solve it. Hint: Use the definition of lim sup; sup; lim inf; inf.

3.Let (an) n in N be a sequence such that lim n->oo (an) = 0: Show that for every ϵ > 0, there exists a subsequence (an(k)) k belong to N such that the series Σ1 {k=1,oo} an(k) converges and |Σ1{k=1,oo} an(k)| < ϵ: Hint: Show that there exists 0 < r < 1 such that Σ1{n=1,oo} rn = r/(1-r) < ϵ: Then construct appropriate (n(k)) k belong to N inductively.

4.Let (an) n in N, (bn) n in N be sequences such that |an+1 – an| < bn

for all n in N and the series Σ1{n=1,oo} bn converges. Show that THE SEQUENCE (an) n in N converges.

5.Dene a function f : R {0} -> R by f(x) =( x^2 – 1)/x

Show that f is continuous on the domain using the ϵ- g denition of continuity. For this problem, it is forbidden to use results on continuity from the textbook and lectures.


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