1
$\begingroup$

I need to know how to exactly evaluate the following sum (in case the limit exists, which is does I think) :

\begin{equation} \lim_{N \to \infty} \quad \sum_{n = 2}^N \frac{1}{n(n-1)} \end{equation}

The reason I am asking is because I would like know wheter I can manipulate expressions involving series by looking at the sequence of partial sums, so is it true in general that

\begin{equation} \lim_{N \to \infty} \quad \sum_{n = 2}^N (a_n - a_{n-1}) = \lim_{N \to \infty} a_N - a_0 \end{equation}

regardless of the limit behaviour of the whether the sequence $(a_n)$ or the series $\lim_{N \to \infty} \sum^N_{n = 2} a_n$ ?

For example, if I let $S_N = \sum_{n = 1}^N = (-1)^{n+1} \frac{1}{n}$ then the resulting series exists. Writing the same partial sum as \begin{equation} T_N = 1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{3} - \frac{1}{6} - \frac{1}{8} + \frac{1}{5} - \frac{1}{10} - \dots \end{equation} (i.e I rearrange the terms) whe have $t_{3N} = \frac{1}{2} s_{2N}$, hence the second series has half the value of the first, although all we did was rearranging the summands before taking the limit.

So this is where my original quest started - I kind of suspect that in order to do arithmetic manipulations before taking the limit I need make sure the sequence $(a_n)$ has certain properties, right ? If yes, what are these properties ?

Many thks for your help!

  • 1
    About whether you can manipulate like that, the answer is no. The answer supplied by Gerry Myerson expected that you would use the hint to find an explicit formula for the sum of the terms from $2$ to $N$ (for which anything is allowed, it is a finite sum) and *then* take the limit.2012-01-22
  • 0
    Ok that makes sense ! I edited the post slightly because I just want to make sure I understand what you say. From what I see you suggest I can always manipulate the partial sum, before taking the limit, without imposing further restrictions on the sequence $(a_n)$. Is that correct ?2012-01-22
  • 0
    Before you take the limit, it's a finite sum, and you can manipulate those to your heart's content.2012-01-22
  • 0
    @harlekin: Your modification is just fine. The equality has to be taken in the sense that the limit on the left exists iff the limit on the right exists, and in that case the two sides are equal.2012-01-22
  • 0
    @Gerry: this makes me a little unsure I understand the example I added to the post above of a series where manipulating the finite sum before taking the limit acutally changes the value of the limit. I took this straight from Victor Bryan's book "Yet another Introduction to Analysis" (page 73). What is the difference between my example and your notion of manitpulating finite sums ? Thks for your help!2012-01-22
  • 0
    Those dots in your $T_N$ mean you're really not talking about a finite sum. If you actually picked some $N$ and wrote out $S_N$, you'd see you can't get $T_N$ as there will be terms missing.2012-01-22

2 Answers 2