∑_(n=1)^∞▒〖(n+1)^3∙((2∙n+2)/(3∙n+1))^n 〗

To find the sum of the series ∑_(n=1)^∞ (n+1)^3 * ((2n+2)/(3n+1))^n, we can use the ratio test to determine if the series converges.

Let a_n = (n+1)^3 * ((2n+2)/(3n+1))^n

We can rewrite the terms of the series as:

a_n = (n+1)^3 * (2(1+1)/(3(1)+1))^n = (n+1)^3 * (4/4)^n = (n+1)^3

Now, let's apply the ratio test:

lim (n→∞) |a_(n+1)/a_n| = lim (n→∞) |(n+2)^3 / (n+1)^3| = lim (n→∞) |(1 + 2/n)^3| = 1

Since the limit is equal to 1, the ratio test is inconclusive. Therefore, we cannot determine if the series converges or diverges using the ratio test.

To find the sum of the series, we can use other methods such as the comparison test, integral test, or other convergence tests.