Math Help - convergence of 1/n as a sequence, but not as a series, why?

1. convergence of 1/n as a sequence, but not as a series, why?

The sequence, 1/n converges to 0 right, because the limit as n-? is 0.

So why does the series of 1/n diverge? Doesn't The sequence's convergence imply that at some point, the series will simply be adding 0's to a finite number?

2. Re: convergence of 1/n as a sequence, but not as a series, why?

Originally Posted by StudentMCCS
The sequence, 1/n converges to 0 right, because the limit as n-? is 0.
So why does the series of 1/n diverge? Doesn't The sequence's convergence imply that at some point, the series will simply be adding 0's to a finite number?
Define the nth partial sum to be ${H_n} = \sum\limits_{k = 1}^n {\frac{1}{k}}$.

Now with induction it is easy to show that $\forall N,~H_{2^N}\ge 1+\frac{N}{2}~.$

Thus $H_n\to\infty~.$