Series Comparison Test for Sum of (1/(sqrt(n^2+1))) and (1/(2n))

[theBEAST]

Homework Statement

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?

 

[DrewD]

note that \sqrt{n^2+1} and 2n 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.

 

[theBEAST]

note that \sqrt{n^2+1} and 2n 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.

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?