Use the ratio test to determine if a series diverges, etc.

Question

does this series (-n+2)^n / ( 3n+4)^n diverge, absolutely converge, or conditionally converge through the ratio test?

Answer 1

The series looks like: 1, 1/7, 0, -1/2197, 1/4096, ...

a_n+1=(-n-1+2)ⁿ⁺¹/(3n+3+4)ⁿ⁺¹=(-n+1)ⁿ⁺¹/(3n+7)ⁿ⁺¹; a_n=(-n+2)ⁿ/(3n+4)ⁿ;

a_n+1/a_n=(-n+1)ⁿ⁺¹(3n+4)ⁿ/[(-n+2)ⁿ(3n+7)ⁿ⁺¹].

a_n+1/a_n→ as (-n)ⁿ⁺¹(3n)ⁿ/[(-n)ⁿ(3n)ⁿ⁺¹]=-n/3n=-⅓ as n→∞. The value of a_n+1/a_n is less than 1 for n, suggesting convergence. Since the ratio of two consecutive term roughly follows the rule a_n+1=-a_n/3, then the sum of two consecutive terms approximates to a_n-a_n/3=⅔a_n, and if we pair consecutive terms such that this sum is positive, i.e., a_2k>a_2k+1 for even n, represented by n=2k. These pairs make up a series of positive terms in which the consecutive term ratio is about ⅔, which is less than 1, implying absolute convergence. Note that the ratio test doesn't work when n=2.

(In fact the sum of the terms converges to about 1.14257.)