MT2002 Analysis

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Exercises 6

  1. For each of the following sequences, decide whether they are eventually monotonic (and prove it).
    (a) ( (n+1)/(n+2) )
    (b) (n + 8/n)
    (c) (n + (-1)n)
    (d) (2n + (-1)n)

    Solution to question 1

  2. Let sn be the partial sum of the series (1/i2). Prove that the sequence (sn) is monotonic and use induction to show that sn≤ 2 - 1/n. Hence prove that the series converges.
    Leonhard Euler (1707 to 1783) showed that its limit is π2/6.

    Solution to question 2

  3. Define a sequence (an) by a1= 1 and an+1= an+ 1/n for n ≥ 1. Prove that the limit as n→ ∞ of |an- an+1| is 0 but that (an) is not a Cauchy sequence.

    Solution to question 3

  4. Let (sn) be the sequence of partial sums of the positive series bn. If 0 ≤ bncn and cn has a convergent sequence of partial sums, deduce that (sn) is monotonic and bounded above and hence convergent.
    This is the so called Comparison Test for positive series discovered by Jacob Bernoulli (1654 to 1705).

    Solution to question 4

  5. Let 0 < r < 1 and let (an) be a sequence for which |an- an+1| < rn for all n. Prove that (an) is a Cauchy sequence.
    Define a sequence (an) by a1= 0, a2= 1 and an+1= an/3 + 2an-1/3 for n > 1. Use the last result to show that this is a Cauchy sequence and hence prove that it is convergent.
    Any guesses about what it converges to?

    Solution to question 5

  6. For any x > 0 we have x0= 1 and 0x= 0. So is the limit as x→ 0+ of xx= 0 or 1 or neither?
    What is the limit as n→ ∞ of n1/n?

    Solution to question 6

  7. If you invest £N at a rate of (say)12% per annum simple interest, then at the end of the year you would have £N(1 + 0.12).
    If the interest were calculated twice (i.e. the rate is 6% per 6 months) you would have £N(1 + 0.06)2 at the end of the year. (This is called compound interest.)
    If the interest were calculated 3 times a year, you would have £N(1 + 0.04)3 at the end. And so on.
    Compounding more and more often, the number of pounds you would have at the end of the year would be the limit as n→ ∞ of the sequence (an) with an= N(1+0.12/n)n.
    Take logarithms, replace 1/n by x and use l'Hôpital's rule to find the limit as x→ 0.

    Solution to question 7

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JOC September 2001