Solutions
an =
10n
(initial value n = 1).
an =
10-n
(initial value n = 1).
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.
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).
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.
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).