find explicit formula for aₙ, find lim n-> ∞ aₙ

Question

a1 = 1/(2-1/2), a2 = 2/(3-1/3), a3 = 3/(4-1/5), a4 = 4/(5-1/5), a5 = 5/(6-1/6).

1) find explicit formula for aₙ

2) find lim n-> ∞ aₙ

Answer 1

desmos.com/calculator/be32fnddrd

Answer 2

an = n/(n+1 - 1/(n+1)) put them over a common denominator, n+1

= n/((n+1)^2/(n+1) - 1/(n+1))

= n/(n^2+2n)/(n+1)

=n(n+1)/(n^2+2n)

=(n^2 +n)/(n^2+2n)

as n approaches infinity, an approaches n^2/n^2 = 1

the highest degree terms dominate