series question!

Question

∑(n=1,∞)arctan(n+1)-arctan(n+2)

Answer 1

∑_n=1^∞[arctan(n + 1) - arctan(n + 2)] = lim n→∞ [arctan2 - arctan3 + arctan3 - arctan4 + ....+ arctan(n) - arctan(n + 1) + arctan(n+ 1) - arctan(n + 2)} = lim n→∞ [arctan2 - arctan(n + 2)] = arctan2 - arctan∞ =

arctan2 - π/2

Answer 2

S_n = sum of first n terms = (Arctan2 - Arctan3) + (Arctan3 - Arctan4) + (Arctan4 - Arctan5) + ... + (Arctan(n+1) - Arctan(n+2))

S_n = Arctan2 - Arctan(n+2)

lim_n→∞ S_n = Arctan2 - π/2

The series converges to Arctan2 - π/2