Show from first principles that (an) is convergent, where an = cos(n) sin (n)/nfor all n ≥ 1. [3 marks](b) Let ℓ, m ∈ R and suppose that (cn) and (dn) are sequences such that(cn + dn) → ℓ + m. Show that (cn) → ℓ if and only if (dn) → m. Anyresults you use must be accurately stated. [4 marks](c) If (xn + yn) is a convergent sequence, must (xn − yn) also be convergent?Give a proof or a counterexample, with justification.

Prove from first principles that the sequence (an) is convergent, wherean = (−1)n + n2n + 1for all n. [5 marks](b) Suppose that a sequence (cn) satisfies |cn| > C|cn−1| for all n ≥ 1, whereC > 1. Show that (|cn|) → ∞. Does it follow that either (cn) → ∞ or(cn) → −∞? Justify your answer. [6 marks](c) If |dn| > |dn−1| for all n, does it follow that (|dn|) → ∞? Justify youranswer with a proof or a counterexample.

Let an = 6n3n + 1.(a)Determine whether {an} is convergent.convergentdivergent    (b)Determine whether ∞n = 1an is convergent.convergentdivergent

Prove from first principles that the sequence (an) is convergent, wherean = (−1)n + n2n + 1for all n.

. Prove, using the ϵ − N definition oflimit, that the sequence (an) given byan = n2n − 1converges to 12 .2. Prove using the definition of limit thatthe sequence (an) given byan = 1(3n − 1)converges to 0.3. Evaluate the following limits(a) limn→∞n4 + 3n1 + n3 + n5 .(b) limn→∞3n − 13n − 3n−1(c) limn→∞√n + 1 − √n(d) limn→∞ nr4 − 1n − 2!4. Let (an) be a sequence. Show that iflimn→∞ an exists, then (an) is bounded.5. Show that the sequence (an) given byan = n2n + 1does not converge.6. Use the sandwich theorem to findlimn→∞1(3n − 1)7. (a) Let n ∈ N. Define functions f, gbyf (x) = (1 + x)nandg(x) = 1 + nxShow by induction that for all nat-ural numbers n ≥ 1 and real num-bers x ≥ −1f (x) ≥ g(x).(b) Sketch graphs f and g case for thecase n = 3.(c) Let a > 1. Show that the sequence(an) is unbounded above.(d) If |a| < 1, show that limn→∞ an = 08. Let x ∈ R. Define a sequence of partialsums by (sn) bys1 = 1s2 = 1 + xs3 = 1 + x + x2... = ...sn = 1 + x + x2 · · · + xn−1Show that (sn) converges if and only if|x| < 1. Find the limit of the sequence(sn) if x = 910 .9. Evaluate:(a) limn→∞1√n(b) limn→∞5n + (−1)n4n(c) limn→∞n34n(d) limn→∞√n√n − 2(e) limn→∞sin n + n3n3(f) limn→∞(−1)n√n1https://pennance.us1

Let (an) and (bn) be bounded sequences, with bn̸ = 0 for all n. Consider thefour sequences(i) (an) (ii) (bn) (iii) (anbn) (iv) (an/bn)(a) Which of the sequences (i)–(iv) have convergent subsequences? Justifyyour answer. [6 marks](b) Suppose also that (an) and (bn) are monotonic. Which of the sequences(i)–(iv) must be convergent? Justify your answer.