Testing a series for convergence/divergence

  • #1

Homework Statement


Use special comparison test to find if [tex]\frac{2+(-1)^n}{n^2+7}[/tex] is convergent or divergent.


Homework Equations


Special comparison test using the convergent series [tex]\frac{1}{n^2}[/tex]

and taking the limit as n-> infinity of my initial series [tex]\frac{2+(-1)^n}{n^2+7}[/tex] divided by my comparison series [tex]\frac{1}{n^2}[/tex]

which comes out to be lim n--> infinity of [tex]\frac{2n^2+(-1)^n(n^2)}{n^2+7}[/tex].


The Attempt at a Solution



I guess I need help evaluating that limit because what I'm getting is undefined (alternating) and the back of the book says that it's defined.
 

Answers and Replies

  • #2
193
2
[tex] 0 \leq \frac{2+(-1)^n}{n^2+7} \leq \frac{3}{n^2+7}. [/tex] You can squeeze it as such. If the series on the right converges, then so does yours.
 

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