Calc analysis - monotone sequences

[Nerpilis]

I think I'm having some trouble on this. First I'll state what the question is then I'll show what i have and my reasoning. Determine if the sequence{b sub(n)} is convergent by deciding on monotonicity and boundness. given:
b sub(n)=n^2/2^n

First I plugged in numbers for n starting with 1. after n=3 the sequence started taking shape and it appears to be approaching zero as n gets large. my feeling at this point is that its monotone decreasing with zero as the lower bound. I guess to prove this i would have to show that b sub(n+1)< b sub(n) thus impling that the lim b sub(n) = 0 or at least exists as n goes to infinity. my first step was to show that b sub(n+1) exists by way of induction. using n=1 in the given i find it true. here is where i start to question myself. b sub(n+1) = (n+2)^2/2^(n+1). I'm not sure how to simplify this but as i plug in numbers I can easily see that 0< b sub(n+1)< b sub(n) therefore leading me to conclude that there is a lim b sub(n) = 0 as n goes to infinity. Am I flawed in this thought process, or are there holes. In my induction part are there ways of simpliying the equation? help and comments are greatly appreciated as i am rusty and many years removed from my last calc class.

 

[EnumaElish]

I think you need to show 0 < b_n+1 < b_n for general n, using induction.

(n+1)^2/2^(n+1) < n^2/2^n
(n+1)^2/n^2 < 2^(n+1)/2^n
n^2 + 2n + 1 < 2n^2
etc...

 

[HallsofIvy]

I'm puzzled by this: " my first step was to show that b sub(n+1) exists by way of induction."

Is there any question that n²/2ⁿ exists for all n?

Since it is also obvious that n²/2ⁿ is positive for all n (so that 0 is a lower bound), to use "monotone convergence" to show that it converges, you need only show that this is a decreasing sequence.

Generally speaking the best way to show that a sequence is decreasing (or increasing) is to compare successive terms. If b_n+1< b_n of all n then we must have b_n+1- b_n< 0 or, alternatively, b_n+1/b_n< 1.

In this case, because b_n is defined as a fraction, I would try the latter of those: b_n+1/b_n= ((n+1)²/2ⁿ⁺¹)(2ⁿ/n²)= (1/2)((n+1)²/n²). Can you show that (n+1)²/n²< 2? Well, that certainly isn't true for n= 1! What n is it true for?