I'm trying to determine the interval of convergence for the series: Using , this simplifies: According to my solution manual, this is incorrect. The limit Where am I going wrong?
Follow Math Help Forum on Facebook and Google+
The calculation should be:
lim |x(n+2)/(n+1)|=|x*lim (n+2)/(n+1)|=|x|*1 For the convergence you need: |x|<1 Now check for x=1 and x=-1
Originally Posted by elim The calculation should be: I know, but I don't see why that's true if I obtain: So I cancel out the Notice that there are TWO factors in the denominator. Where does that second factor go?
Originally Posted by Also sprach Zarathustra lim |x(n+2)/(n+1)|=|x*lim (n+2)/(n+1)|=|x|*1 For the convergence you need: |x|<1 Now check for x=1 and x=-1 I understand from that point, my problem is the simplification of . See my last reply to elim
Originally Posted by adkinsjr I know, but I don't see why that's true if I obtain: So I cancel out the Notice that there are TWO factors in the denominator. Where does that second factor go? The first equality is incorrect. How did you get in the numerator? It should be - you have an extra factorial there.
I think I've got it now. The sequence can be written as That seems right.
Originally Posted by Defunkt The first equality is incorrect. How did you get in the numerator? It should be - you have an extra factorial there. I don't see what you mean. The numerators are both the same (as far as I can tell), just rearranged.
Oh, you're right; my bad. Looks like you just missed the term from the numerator of the first fraction.
The first thing I would do is simplify to . The series you have is simply .
hmmm.... I don't know about this simplification...
Originally Posted by Also sprach Zarathustra hmmm.... I don't know about this simplification... What do you not understand about that? It is completely obvious.
Maybe it because of the night, I don't know! Anyway... {2*4*6...*(n+1)}*{1*3*5*7*...*n} =(n+1)! Edit: ooops(again!) I have not noticed that is not only the even numbers! Sorry!
Originally Posted by HallsofIvy The first thing I would do is simplify to . The series you have is simply . Yes, I realize I should have done that. I was foolishly jumping right into the ratio test, without looking for any simplifications in the series itself. I wasted too much time on this, lol.
Originally Posted by Also sprach Zarathustra Maybe it because of the night, I don't know! Anyway... {2*4*6...*(n+1)}*{1*3*5*7*...*n} =(n+1)! Edit: ooops(again!) I have not noticed that is not only the even numbers! Sorry! A product of even numbers can be reduced as: . A product of odd numbers is more complicated- you have to put the even numbers in: .
View Tag Cloud