Analyze convergence of the series $\sum\limits_{n=1}^\infty\frac{1}{n(n^\frac14-n^\frac13+n^\frac12)}$ $$\sum_{n=1}^\infty\frac{1}{n(n^\frac{1}{4}-n^\frac{1}{3}+n^\frac{1}{2})}$$
It is possible to use the ratio test or the root test or any of them without give opposite answers ? There is a case in which a method give an answer and another other answer? Which of them is a better strategy for this one? Does this one converge or diverge? How I recognize which method I ought use?
Sorry for asking so many questions.

Just for the fun.
Consider $$A=\frac{1}{x(x^\frac{1}{4}-x^\frac{1}{3}+x^\frac{1}{2})}$$ and make $x=y^{12}$ to get $$A=\frac{1}{y^{15} \left(y^3-y+1\right)}=\frac{1}{y^{18} \left(1-\frac 1{y^2}+\frac 1 {y^3}\right)}$$ Now, make the long division (or use Taylor series) to get $$\frac{1}{ 1-\frac 1{y^2}+\frac 1 {y^3}}=1+\frac 1{y^2}-\frac 1 {y^3}+\frac 1 {y^3}+\frac 1 {y^4}+O\left(\frac 1 {y^5} \right)$$ making $$A=\frac 1 {y^{18}}+\frac 1 {y^{20}}-\frac 1 {y^{21}}+\frac 1 {y^{22}}+O\left(\frac 1 {y^{23}} \right)$$ Back to $x$ $$A=\frac 1 {x^{3/2}}+\frac 1 {x^{5/3}}-\frac 1 {x^{7/4}}+\frac 1 {x^{11/6}}+\cdots$$ Replace $x$ by $n$ and prepare the summations : you face convergent series.
