2011-04-06, 10:22 PM
I'm doing the basics of sequences like converging, diverging, monotonic or not, upper/lower bounds and I need help.
Determine if the sequence converges or diverges.
1. a[SIZE="1"]n[/SIZE] = 1/n^2 + 2/n^2 + ... + n/n^2
My first thought was to multiply top and bottom by 1/n^2 to put all the n's on the bottom but that gave me 0 instead of 1/2, so my approach is probably wrong.
Getting that wrong shook my confidence, so I'd appreciate it if you guys can check my other answers, especially my reasoning/explanation since my teacher makes a big deal about that.
2. a[SIZE="1"]n[/SIZE] = (-1)^n sin (1/n)
This one I was pretty unsure of. What I did was write statements for two cases (n is even/odd). Since they both go to 0 as n-> infinity , the sequence converges to 0.
3. a[SIZE="1"]n[/SIZE] = ln (n+1) - ln n
First, I set f(x) = y = ln ((x+1)/x). Then I did e^y = x+1/x. Taking the limit x-> infinity for both sides, I got e^y -> 1, so y -> ln 1 = 0.
Therefore, lim a[SIZE="1"]n[/SIZE] as n->infinity is 0 and a[SIZE="1"]n[/SIZE] converges to 0.
3. a[SIZE="1"]n[/SIZE] = (1 + 3n)^ (1/n)
First, I set f(x) = y = (1 + 3x)^ (1/x). Then I did ln y = (1/x) ln (1+3x). Taking the limits of both sides again, I get ln y -> 0 and so y -> e^0 = 1.
Therefore, a[SIZE="1"]n[/SIZE] converges to 1
Additionally, there is a problem on the bounds of a sequence.
4. Find the least upper bound and the greatest lower bound of a[SIZE="1"]n[/SIZE] = (e^n)/ (n+1)!
The sequence is decreasing, so I think the least upper bound would simply be a[SIZE="1"]1[/SIZE]. I don't know what to do for the lower bound.
Determine if the sequence converges or diverges.
1. a[SIZE="1"]n[/SIZE] = 1/n^2 + 2/n^2 + ... + n/n^2
My first thought was to multiply top and bottom by 1/n^2 to put all the n's on the bottom but that gave me 0 instead of 1/2, so my approach is probably wrong.
Getting that wrong shook my confidence, so I'd appreciate it if you guys can check my other answers, especially my reasoning/explanation since my teacher makes a big deal about that.
2. a[SIZE="1"]n[/SIZE] = (-1)^n sin (1/n)
This one I was pretty unsure of. What I did was write statements for two cases (n is even/odd). Since they both go to 0 as n-> infinity , the sequence converges to 0.
3. a[SIZE="1"]n[/SIZE] = ln (n+1) - ln n
First, I set f(x) = y = ln ((x+1)/x). Then I did e^y = x+1/x. Taking the limit x-> infinity for both sides, I got e^y -> 1, so y -> ln 1 = 0.
Therefore, lim a[SIZE="1"]n[/SIZE] as n->infinity is 0 and a[SIZE="1"]n[/SIZE] converges to 0.
3. a[SIZE="1"]n[/SIZE] = (1 + 3n)^ (1/n)
First, I set f(x) = y = (1 + 3x)^ (1/x). Then I did ln y = (1/x) ln (1+3x). Taking the limits of both sides again, I get ln y -> 0 and so y -> e^0 = 1.
Therefore, a[SIZE="1"]n[/SIZE] converges to 1
Additionally, there is a problem on the bounds of a sequence.
4. Find the least upper bound and the greatest lower bound of a[SIZE="1"]n[/SIZE] = (e^n)/ (n+1)!
The sequence is decreasing, so I think the least upper bound would simply be a[SIZE="1"]1[/SIZE]. I don't know what to do for the lower bound.