- #1
NastyAccident
- 61
- 0
Homework Statement
[tex]\sum^{\infty}_{n=1}\frac{3}{n^{1+\frac{1}{n}}}[/tex]
Homework Equations
Comparison Test
Limit Comparison Test
Test of Convergence (Just to show it doesn't immediately diverge)
The Attempt at a Solution
I sort of just would like to check to make sure I'm getting a proper [tex]b_{n}[/tex]
Manipulating the Series:
[tex]\sum^{\infty}_{n=1}\frac{3}{n^{1+\frac{1}{n}}}[/tex]
[tex]\sum^{\infty}_{n=1}\frac{3}{n^{\frac{n+1}{n}}}[/tex]
[tex]\sum^{\infty}_{n=1}\frac{3}{n\sqrt[n]{n}}[/tex]
Test of Convergence:
limit n->infinity [tex]\frac{3}{n^{\frac{n+1}{n}}}[/tex]
limit n->infinity [tex]\frac{3}{n^1}}[/tex]
limit ->infinity [tex]0[/tex]
The series MAY or MAY NOT be convergent.
Comparison Test
*Note* This series only contains positive terms*
From the looks of it, I'm going to GUESS that this series DIVERGES.
[tex]a_{n} = \frac{3}{n^{1+\frac{1}{n}}}, b_{n} = \frac{1}{n}[/tex]
[tex]a_{n} \geq b_{n}[/tex]
Since the series is [tex]\sum^{\infty}_{n=1} \frac{1}{n}[/tex], it is a p-series and it diverges because p [tex]\leq[/tex] 1.
By the Comparison Test, [tex]\sum^{\infty}_{n=1}\frac{3}{n\sqrt[n]{n}}[/tex] also diverges.
Limit Comparison Test
From the looks of it, I'm going to GUESS that this series DIVERGES.
*Note* This series only contains positive terms*
[tex]a_{n} = \frac{3}{n^{1+\frac{1}{n}}}, b_{n} = \frac{1}{n}[/tex]
limit n->infinity [tex]\frac{\frac{3}{n^{\frac{n+1}{n}}}}{\frac{1}{n}}[/tex]
limit n->infinity [tex]\frac{3n}{n^{\frac{n+1}{n}}}}[/tex]
limit n->infinity [tex]\frac{3}{n^{\frac{1}{n}}}}[/tex]
limit n->infinity [tex]0[/tex]By the Limit Comparison Test, [tex]\sum^{\infty}_{n=1}\frac{3}{n\sqrt[n]{n}}[/tex] is divergent since 0 > 0.
My questions:
Did I pick the right [tex]b_{n}[/tex]? If not, what did I do wrong in picking [tex]b_{n}[/tex]?
Any hints for picking the proper [tex]b_{n}[/tex]?
Was there a step that I missed or was unclear?
As always, any and all help is appreciated and will be greatly thanked! =) (I'm getting a 96% in Calc II thanks to the help I am receiving from this community in understanding concepts! [Nailed a 56/60 on a 20% exam!])
NastyAccident.