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Homework Statement
I am studying for a calculus test tomorrow on this website (http://archives.math.utk.edu/visual.calculus/6/index.html). I am working on the limit comparison test problems but I am unfamiliar with the form they use in their solutions. For example:
Limit comparison test (prove convergence / divergence)
Series (from n=1 to ∞) 1/(sqrt(n)+sqrt(n+1))
Homework Equations
The Attempt at a Solution
Series (from n=1 to ∞) 1/(sqrt(n)+sqrt(n+1)) compare to Series (from n=1 to ∞) 1/sqrt(n), which we know diverges by the p-series test (p=.5, p<1)
Let an = 1/(sqrt(n)+sqrt(n+1)) and bn = 1/sqrt(n)
lim(n->∞) an/bn
= lim(n->∞) 1/(sqrt(n)+sqrt(n+1)) / 1/sqrt(n)
= lim(n->∞) sqrt(n)/(sqrt(n)+sqrt(n+1))
I can see why this limit is 1/2 but I don't understand how the solution to the problem ended up in the form
lim(n->∞) 1/(1+sqrt(1-1/n)) = 1/2
I can tell by the other solutions that this form is a rule of sorts but I don't know what it is.
Thank you for the help.