# Sequences and series: exercise sheet 4

Exercise.

Consider the sequence defined by 1=1 and +1= 1+ .

(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 1 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 ()= 1+ (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).

Exercise.

Find and where = 1 -1 , ,1 .

Exercise.

For the following series series =1 , determine if the series converges and if so find the limit. (Hint: use partial fractions to express .)

(c) =1 ( 2-1) (As 1 is not defined consider =2 .)

Exercise.

(a) Let

0

in and . Prove that (1+

) 1+

+ (-1) 2

2

. (Use induction on .)

(b) If with 0 1 , show that 0 as . (Hint: write =11+

and use (a).)

(c) Let = where 0 1 . Show that =1 = (1-)2 (1-(+1) + +1 ) (Use induction on .)

(d) Hence show that the series =1 converges if 0 1 , and find its limit.

(e) Show that =1 diverges if 1 .

Standard results and theorems, including the squeeze rule, may be used if quoted correctly.

## Sample solutions to selected exercises

### Exercise 4.3(a)

Let = =1 .

Note that =1(+1) =1 -1+1 so =11-12+12-13+13-14++ 1 -1+1 hence =11-1+1 for all . But 1+1 0 as (being a subsequence of a standard null sequence 1 ) and so 1-0=1 by continuity of - and therefore the series converges with =1 =1 .

### Exercise 4.4(b)

Since 0 1 there is a real number

such that =11+

and

0

. Now note that 0 = (1+

)

1+

+ (-1) 2

2

by part (a). Simplifying this we have 0 2 2 2 +2

+

2

-

2

. The sequence ( ) on the right hand side of this inequality converges to 0 by the standard null sequence 1 and continuity of addition, subtraction, multiplication, and of division at (0,

2)

. (Note that

20

.) Thus by the squeeze rule applied to 0 0 we have 0 as as required.