(a) ∑4/[n^(1+(1/n))] for n∈[4,∞)=4∑1/[n^(1+(1/n))] for n∈[4,∞).
So, expanding:
4[1/4¹˙²⁵+1/5¹˙²+1/6⁷⸍⁶+...]
Ratio of the first two terms in parentheses is:
4¹˙²⁵/5¹˙²=0.82 approx.
Ratio of the 10th and 11th terms (for example) is:
7^(8/7)/8^(9/8)=0.89 approx.
As n gets larger the ratio approaches 1.
This strongly suggests that the series is divergent. To be convergent the ratio needs to be less than 1.
(b) ∑e⁵⸍ⁿ/n for n∈[1,∞).
The ratio of consecutive terms is:
(n/(n+1))e^(-5/(n²+n)), because 5(1/(n+1)-1/n)=-5/(n²+n).
For large n, the exponential approaches e⁰=1 and n/(n+1) also approaches 1.
The ratio approaches 1 as n→∞, so the series appears to be divergent.
(c) ∑8/√(n²+4) for n∈[1,∞)=8∑1/√(n²+4) for n∈[1,∞).
As n gets larger √(n²+4)→n so the ratio of consecutive terms approaches n/(n+1) which in turn approaches 1, implying divergence.