Question

40 points!! What is the limit of the infinite series?

Answer 1

As n goes onto infinity, the expression
`(3n^5)/(4n^5+1)` will approach
`(3)/(4)`. Note the 3 and 4 come from the leading coefficients of the numerator and denominator respectively. This only works because the degrees for the numerator and denominator are both the same (both are 5).

Since
`(3n^5)/(4n^5+1)` is not approaching 0 when n gets really large, adding on successive terms of these values will have the infinite sum diverge. After some very large n, we are effectively adding on 3/4 each time and that just makes the overall sum keep growing forever.

Answer: The sum diverges