Question

using root test solve summation as n=1 goes to infinity of (-1)^n (ln (n)/n)^n

Answer 1

∑_{n=1}^{∞}(-1)ⁿ(((ln n)/n))ⁿ
Root test: study the limit
lim_{n→∞}[n]√(|(-1)ⁿ(((ln n)/n))ⁿ|)=lim_{n→∞}((ln n)/n)=0<1
⇒ the series converges
"solve summation" sounds innapropriate ...
more likely "study the convergence"
as the root test doesn't allow you to find the value of the sum... it only gives you qualitative information about convergence/divergence