Series Convergence

  • #1
Given [tex]a_{n} > 0[/tex] and [tex]\sum a_{n}[/tex] diverges, show that [tex]\sum \frac{a_{n}}{1+a_{n}}[/tex] diverges.
Since I don't have an explicit form for the series, I can't apply any of the standard tests. I'm not sure where to start on this problem. I know the criteria for convergence/divergence, namely tail end of series has to converge or cauchy criterion condition. But I don't see how that helps without knowing what series looks like. Please steer me in the right direction.
 

Answers and Replies

  • #2
Fermat
Homework Helper
872
1
If a series diverges, what happens to to its reciprocal ?
 
  • #3
I would say reciprocal converges, but apparently it's not enough to bring original series to convergence... I thought about this a little more and I think I'll analyze it based on how [tex]a_{n}[/tex] diverges. that is, does it go to zero, constant, or infinity as n goes to infinity and try to bound the reciprocal from below to show series diverges.
 
  • #4
665
0
For [itex]a_n>1[/itex]:

[tex]\frac{a_n}{1+a_n}>\frac{1}{2}[/tex]

For [itex]a_n\leq 1[/itex]:

[tex]\frac{a_n}{1+a_n}\geq\frac{a_n}{2}[/tex]
 

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