- #1
Nerpilis
- 19
- 0
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.
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.