Question

test for convergence or divergence:
inf(sigma)n=1 [(5n-3)/(n^2-2n+5)]

Answer 1

It converges. You can replace all of that with (for convergence tests only)\[\sum_{n=1}^{\infty}(1/(n^2)\] which converges according to the p-test.

Convergent :D

Answer 2

It is divergent. Use the limit comparison test! compare to \(b_n=\frac{1}{n}\).
The limit of \(a_n/b_n\) is 5 but \(\sigma b_n\) is divergent. So the original is divergent as well.

Answer 3

Why would you compare it to 1/n?

Answer 4

Because the degree difference between the top and the bottom is 1. So roughly the series behave like the harmonic series.

Answer 5

Good answer. Wasn't thinking :D