Question

∑from1 to ∞((6^n +5)/(6^n+1)

Answer 1

lim(n->∞) (6^n +5)/(6^n+1) = 1
so I doubt the series converges

Answer 2

Are you using the diverges test? How did you got 1?

Answer 3

as n gets huge, the terms approach 6^n/6^n = 1 (6^n is so huge that the +5 and +1 make no difference)
for any series to converge, the terms of the sequence need to go to zero.
These terms approach 1, so the series becomes
...+ 1+1+1+1...
no way that can converge

you can also try the ratio test. The ratio of two successive terms must be less than 1. Yours is exactly 1.