Difference between $\lim_\limits{n\rightarrow\infty}\sum_\limits{k=1}^{n}a_k$ and $\sum_\limits{k=1}^{\infty}a_k$

Question

Is there any difference between $\lim_{n\rightarrow\infty}\sum_\limits{k=1}^{n}a_k$ and $\sum_\limits{k=1}^{\infty}a_k$?

My example and thought:

Let $a_n=n$ where $n\in\mathbb{P}$. $\mathbb{P}$ is the set of all positive integer.

Then $$\lim_{n\rightarrow\infty}\sum_{k=1}^{n}a_k=\lim_{n\rightarrow\infty}\sum_{k=1}^{n}k=\lim_{n\rightarrow\infty}\frac{n(n+1)}{2}\rightarrow\infty$$

And

$$\sum_{k=1}^{\infty}a_k=\sum_{k=1}^{\infty}k=-\frac{1}{12}.$$

Am I thinking right?

Answer 1

Infinite sums are defined as limits of their partial sums. So in this sense they are equivalent.

What you are using in your example is a Zeta regularization of a divergent series. In that sense, even the equality sign is not really justified. $-1/12$ is not the sum in the traditional sense, it's a generalization and you have to specify what you are doing.

Answer 2

Your example is a red herring; the standard way of interpreting the sum $\sum_{k=1}^\infty k$ is that it diverges to infinity, although there are some theories of divergent series which would assign the sum a different value. In any case, the two limits in your question are exactly equal: an infinite series is defined to be the limit of its partial sums.