I am maybe overthinking this, but what is the lim as n-> infinity of |(n+1)/(n+2)|

Question

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I am trying to use the ratio test to find interval of convergence for the infiinite series (n/n+1)((-2x)^(n-1))
so i did the limit of |(n+1)/(n+2)| but i am wondering if the limit is simply =1, or if it is infinity/infinity so use l'hopital's rule???
maybe i am overthinking it and the limit is just 1.
thank you for your assistance

Answer 1

the limit of your problem is 1
as n --> infinity , both numerators and denominator are practiacally the same, with the denominator being one less than the numerator.
e.g. if n = 1 000 000
then we have 1 000 001/1 000 002 , pretty close to 1 wouldn't you say?

Answer 2

since (n+1)/(n+2) = 1 - 1/(n+2) it is clear that as n grows, the value approaches 1.

Yes, L'Hopital's Rule also works, since you have infinity/infinity. taking derivatives gives 1/1 as the limit.