Solutions
  1. an = 10n  (initial value n = 1).
     
  2. an = 10-n  (initial value n = 1).
     
  3. The series converges against 1. The n-th item may be written as

    (1+3/n)/(1+4/n).

    As n increases above any bound, the terms 3/n and 4/n decrease and - after a sufficient number of steps - fall below any given positive number. (In other words: when interpreted as series, they converge against 0). Since performing the limit commutes with addition and multiplication (as long there is no danger of explosion from the denominator), these terms may be omitted, and what remains as the limit of the given series is just 1/1, hence 1.
     
  4. From the 32nd. The program shows that

    a31 = 2.001008064516129

    (here, the inequality is not yet satisfied), and

    a32 = 2.0009469696969697

    (here and for all succeeding items, the inequality holds).
     
  5. an takes values between -1 and 1, and does not converge.
    (Interesting additional exercise: Closer inspection of the series suggests that at most four succeeding items carry the same sign. Can you provide a reason for this? Hint: look at the graph of the sine function. The explanation is related to the number p).

    bn converges against 0, because the numerators are trapped between -1 and 1, whereas the denominators increase above any bound for large n.
     
  6. The code defines a sequence in a recursive way: the first two items are 1, all others are the sum of the two preceeding ones. It is called Fibonacci sequence. Its items are called Fibonacci numbers and often appear in natural patterns (e.g. in sun flowers, on snail's shells and in the molecular structure of nerve cells).