Does the following series converge or diverge? Justify.

Question

\sum _{n=2}^{\infty }\:\frac{1}{\sqrt{n^3-2n^2+5n-7}}

Answer 1

∑_{n=2 to ∞} [1 / √(n³ - 2n² + 5n - 7]

Try the limit comparison test with a_n = 1 / √(n³ - n² + 5n - 7) and b_n = 1 / √n³ = 1/n^3/2

∑ _{(n=2 to∞)} b_n converges since it is the tail end of the p-series with p = 3/2 > 1

a_n / b_n = √ [n³ / (n³ - n² + 5n - 7)] = √ [ 1 / (1 - 1/n + 5/n² - 7/n³)]

lim_n→∞ [a_n / b_n ] = 1, so ∑_{(n=2 to ∞)} a_n also converges.

Answer 2

I think you are asking about the sum from n=2 to ∞ of 1/[sqrt(n³ -2n² + 5n -7)].

I cannot answer algebraically but graphically, it appears that the terms of the required sum are less than 1/x^1.1 for x > 2.35. This means the sum converges by the comparison test.