Question

Determine whether the sequence:

ln(2n^2 +1) - ln(n^2 +1)

converges or diverges. If the sequence converges, find the limit.

Answer 1

`\\ \lim_(n \to \infty) ln( (2n^2+1) )-ln((n^2+1)) \\ \\ \lim_(n \to \infty) \ln{ (2n^2+1)/(n^2+1) } \\ \\ \lim_(n \to \infty) \ln{ ( (2n^2)/(n^2) + (1)/(n^2))/( (n^2)/(n^2)+ (1)/(n^2)) } \\ \\ \lim_(n \to \infty) \ln{ ( 2 + (1)/(n^2))/( 1+ (1)/(n^2)) } \\ \\ \ln{ ( 2 + 0)/( 1+ 0) } \\ \\ ln(2)`