Does the series $\sum{(\frac{1}{n^2} - \frac{1}{n})}$ converge or diverge?

Question

Whats about the given series, Converge or Diverge? The series is given $$ \sum_1^\infty{(\frac{1}{n^2} - \frac{1}{n})} $$

Answer 1

Hint: $\sum \frac{1}{n^2}$ converges absolutely and $\sum \frac{1}{n}$ diverges, so what about the sum?

Answer 2

If the series were convergent assume the sum is $l$. This would mean all the terms being positive that

$$\sum_{n=1}^\infty{1\over n}=-l+{\pi^2\over 6}$$

But the harmonic series is divergent so the initial series is divergent

Answer 3

$$\sum\limits_{n=1}^{\infty}{(\frac{1}{n^2}-\frac{1}{n})}:=\lim\limits_{M\to\infty}{\sum\limits_{n=1}^{M}{(\frac{1}{n^2}-\frac{1}{n})}}=\lim\limits_{M\to\infty}{\sum\limits_{n=1}^{M}{\frac{1}{n^2}}}-\lim\limits_{M\to\infty}{\sum\limits_{n=1}^{M}{\frac{1}{n}}}=:\sum\limits_{n=1}^{\infty}{\frac{1}{n^2}}-\sum\limits_{n=1}^{\infty}{\frac{1}{n}}$$ And you know that $\sum\limits_{n=1}^{\infty}{\frac{1}{n}}$ diverges. So the initial series also diverges.