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Series Comparison Test

  1. Mar 18, 2012 #1
    1. The problem statement, all variables and given/known data
    For the sum from n=1 to ∞ (1/(sqrt(n^2+1)), I know you can use the limit comparison test to show that it is divergent but I was wondering if it is possible to compare this with 1/(2n)? I am not sure if 1/(2n) is always less than (1/(sqrt(n^2+1)) within those bounds. How could I show that it always is?
     
  2. jcsd
  3. Mar 18, 2012 #2
    note that [itex]\sqrt{n^2+1}[/itex] and [itex]2n[/itex] are positive here, so you don't have to worry about either one changing the direction of an inequality. Set up the inequality and do some algebra.
     
  4. Mar 18, 2012 #3
    Okay so I found that the inequality is true when n > sqrt(1/3), since our bounds are n>1 then it is always true for this sum. Thus I can use that to compare and prove that 1/(sqrt(n^2+1)) is divergent correct?
     
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