Determine the convergence of ∑n=1∞1n1+1/n\sum_{n=1}^{\infty} \frac{1}{n^{1 + 1/n}}∑n=1∞n1+1/n1.
Which test is most direct?
By Limit Comparison Test with ∑1n\sum \frac{1}{n}∑n1: the series diverges.
By Limit Comparison Test with ∑1n\sum \frac{1}{n}∑n1: the series converges.
By the Root Test: 1/n1+1/nn=n−(1/n2+1/n3)\sqrt[n]{1/n^{1+1/n}} = n^{-(1/n^2 + 1/n^3)}n1/n1+1/n=n−(1/n2+1/n3), which has limit 111, making the test inconclusive.
By noting that 1+1/n→11 + 1/n \to 11+1/n→1, we compare to ∑1n\sum \frac{1}{n}∑n1, which diverges by symmetry.