Question

Hi, just wanted to verify my work please.

Determine if the sequence {(n+1)^2/n^2+1} is increasing or decreasing and find it's lower and/or it's upper bound.

First I have n>n+1, therefore it's decreasing.

Then, if I do the limit n->inf, I get = 1, so 1 will be my lower bound.
Since it's decreasing, my first term of a1 when n=1 gives 2, so my upper bound is 2.

Is this correct?

Thank you

Answer 1

I assume you are saying
term(n) = (n+1)^ / (n^2 + 1)
= (n^2 + 2n + 1)/(n^2 + 1)

what about limit (n^2 + 2n + 1)/(n^2 + 1) as n ---> ∞
= lim (1 + 2/n + 1/n^2) / (1 + 1/n^2)
= (1+0+0)/(1+0)
= 1

investigate
term(4) = 25/17 >1
term(5) = 36/26 > 1 but term(5) < term 4)
so the terms are getting smaller and approach 1

you had that, and assuming that n is a natural number
term(1) would indeed be 4/2 = 2

so I agree

I don't like your statement : n>n+1 , it is a false statement, since it says 0 > 1
you meant: term(n) > term(n+1)