- #1
bz89
- 1
- 0
Homework Statement
Consider the sequence {an} where a1 = sqrt(k), an+1 = sqrt(k + an), and k > 0.
a. Show that {an} is increasing and bounded.
b. Prove that the limit as n approaches infinity of an exists.
c. Find the limit as n approaches infinity of an.
The attempt at a solution
b is straightforward. If you show that an is monotonic and bounded then it has a limit.
I don't really understood how to approach a. The solutions guide suggests some sort of induction that starts with an <= ((1 + sqrt(1 + 4k))/2). I don't understand how I would be able to go from the givens to that point.
Consider the sequence {an} where a1 = sqrt(k), an+1 = sqrt(k + an), and k > 0.
a. Show that {an} is increasing and bounded.
b. Prove that the limit as n approaches infinity of an exists.
c. Find the limit as n approaches infinity of an.
The attempt at a solution
b is straightforward. If you show that an is monotonic and bounded then it has a limit.
I don't really understood how to approach a. The solutions guide suggests some sort of induction that starts with an <= ((1 + sqrt(1 + 4k))/2). I don't understand how I would be able to go from the givens to that point.