Is $\sum_{n=1}^{\infty} \frac{n-1}{n^2}$ convergent or divergent? Is $\sum_{n=1}^{\infty} \frac{n-1}{n^2}$ convergent or divergent?
I tried ratio test but didn't seem to work, and I also know that the limit goes to zero, but I can't say its convergence because then.
Can someone show me a test to see if its convergent or divergent?
 A: Hint: Use limit comparison test with $b_n = \frac{1}{n}$. 
A: Using Cauchy condensation test, its convergence is equivalent to
$$ \sum_{n=1}^\infty 2^n \frac{2^n - 1}{2^{2n}} = \sum_{n=1}^\infty \frac{2^n - 1}{2^n}
$$
so...
A: As commented, $\frac{n-1}{n^2}=\frac{1}{n}-\frac{1}{n^2}$. Therefore,
$$\sum_{n=1}^N\frac{n-1}{n^2}+\sum_{n=1}^N\frac{1}{n^2}=\sum_{n=1}^N\frac{1}{n}.$$
If your proposed series were convergent, then the left side would converge, and hence the right side would too, a contradiction.
A: Another possible way to show the divergence :
For $n > 2$, you have that $n-1\geq \frac{n}{2}$ and that imply that $\frac{n-1}{n^2} \geq \frac{1}{2}\frac{1}{n}$, but $\sum \frac{1}{n}$ diverge, so $\sum_{n-1}{n^2}$ will diverge too
A: It seems like a few people have pointed you in the right direction, but you still need help.
The limit comparison test.  If you have two series $\sum a_n$ and $\sum b_n$ and you know what whether a_n converges or diverges.
And,
$\lim_\limits{n\to \infty} \dfrac{b_n}{a_n} = M$ for some bounded M not equal to $0.$ Then $b_n$ converges when $a_n$ converges and diverges when $a_n$ diverges.
And this one I like but is slightly stupid.
the direct comparison test.... $\frac{n-1}{n^2} < \frac{1}{n}$ so you can't run the direct comparison test vs. $\frac{1}{n}$.  However   $\frac{n-1}{n^2} > \frac{1}{2n}$ for all $n>2$.
A: Try it this way:  $\sum_{n=1}^{\infty} \frac{1}{n^2}$ definitely converges, right?
Now if your series $\sum_{n=1}^{\infty} \frac{n-1}{n^2}$ converges, then the sum $\sum_{n=1}^{\infty} \frac{n-1}{n^2} + \sum_{n=1}^{\infty} \frac{1}{n^2}$ would converge as well.
But that sum is $\sum_{n=1}^{\infty} \frac{n}{n^2} = \sum_{n=1}^{\infty} \frac{1}{n}$.  Does it converge?
