Consider the sequence defined by and .
(a) Assuming for the moment that is
convergent, by solving a quadratic equation or otherwise
make a reasonable
conjecture for a value
that might be the limit of the sequence.
(b) Prove that holds for all , by induction. Your proof must not use the conjecture from (a) that , only the value of itself.
(c) Using part (b), or by induction, prove that the sequence is monotonic nondecreasing.
(d) Which theorem from lectures allows you to deduce from (b) and (c) that has a limit?
(e) Apply the continuity of the function (considered as a function of positive into the set of positive reals) to prove that the limit of the sequence is , as conjectured in part (a).
Find and where .
For the following series series , determine if the series converges and if so find the limit. (Hint: use partial fractions to express .)
(c) (As is not defined consider .)
(a) Let in and . Prove that . (Use induction on .)
(b) If with , show that as . (Hint: write and use (a).)
(c) Let where . Show that (Use induction on .)
(d) Hence show that the series converges if , and find its limit.
(e) Show that diverges if .
Standard results and theorems, including the squeeze rule, may be used if quoted correctly.
Note that so hence for all . But as (being a subsequence of a standard null sequence ) and so by continuity of and therefore the series converges with .
Since there is a real number such that and . Now note that by part (a). Simplifying this we have . The sequence on the right hand side of this inequality converges to by the standard null sequence and continuity of addition, subtraction, multiplication, and of division at . (Note that .) Thus by the squeeze rule applied to we have as as required.