Question

use the comparison test to determine the convergence or divergence of the series:
4 + 1/5 + .3 + 1/(3+√2) + 1/(9+√3) + 1/(27+√4)... showing all work.

I'm stuck on this one.

Answer 1

neglect the first 3 terms, the series is
1/[ 3^n + √(n+1) ], where n greater than or equal to 1

1 / [3^n + √(n+1) ] < 1/3^n = (1/3)^n

but (1/3)^n is an geometric series with |r| = |1/3| < 1, so it is convergent. Therefor, the series 1/[3^n + √(n+1)] is also convergent

a convergent series that is added to a finite number of terms (in this case the first 3 terms) is also convergent.

Answer 2

I've spent thirty minutes trying to decipher the first terms, thanks a ton!