Use the Limit Comparison Test to determine if ∑n=1∞n2+1n5−n+3\sum_{n=1}^{\infty} \frac{n^2 + 1}{n^5 - n + 3}∑n=1∞n5−n+3n2+1 converges or diverges.
Converges; compare to ∑1n3\sum \frac{1}{n^3}∑n31 with limit L=1L = 1L=1
Diverges; compare to ∑1n\sum \frac{1}{n}∑n1 with limit L=∞L = \inftyL=∞
Converges; compare to ∑1n2\sum \frac{1}{n^2}∑n21 with limit L=0L = 0L=0
Diverges; the Limit Comparison Test is inconclusive