1. The problem statement, all variables and given/known data
$$\sum\limits_{n=1}^∞ \frac{1}{n√(n)} $$
Since $$ \frac{1}{n√(n)} \equiv \frac{1}{x^{3/2}} $$ this is a convergent p-series. But, when I attempt to prove this by the limit comparison test with known convergent series such as $$\sum\limits_{n=1}^∞ \frac{1}{n^2}$$
ex. $$\lim_{n \to ∞} \frac{\frac{1}{n√(n)}}{\frac{1}{n^2}} = ∞ $$
The limit comparison test does not prove this fact. But I get a real number N when comparing the known divergent series $$\sum\limits_{n=1}^∞ \frac{1}{n}$$ with $$\sum\limits_{n=1}^∞ \frac{1}{n√(n)} $$
ex. $$\lim_{n \to ∞} \frac{\frac{1}{n√(n)}}{\frac{1}{n}} = 0 $$ Can someone please explain the reasoning behind this?
$$\sum\limits_{n=1}^∞ \frac{1}{n√(n)} $$
Since $$ \frac{1}{n√(n)} \equiv \frac{1}{x^{3/2}} $$ this is a convergent p-series. But, when I attempt to prove this by the limit comparison test with known convergent series such as $$\sum\limits_{n=1}^∞ \frac{1}{n^2}$$
ex. $$\lim_{n \to ∞} \frac{\frac{1}{n√(n)}}{\frac{1}{n^2}} = ∞ $$
The limit comparison test does not prove this fact. But I get a real number N when comparing the known divergent series $$\sum\limits_{n=1}^∞ \frac{1}{n}$$ with $$\sum\limits_{n=1}^∞ \frac{1}{n√(n)} $$
ex. $$\lim_{n \to ∞} \frac{\frac{1}{n√(n)}}{\frac{1}{n}} = 0 $$ Can someone please explain the reasoning behind this?
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