Why not to calculate the limit of individual terms when the number of terms approach infinity due to limiting variable?

Question

My instructor was using this example to illustrate his point-

$$ \lim_{n\to \infty} \frac{1+2+3+...+n}{n^2}$$ The numerator is an AP and the limit can be easily calculated this way to be a finite number($0.5$ in this case).

But, if we use the sum law and separate all the $\frac{r}{n^2}$ terms, the individual limits will be $0$ and so will be the answer. This is what my instructor told me to avoid.

My question is-

Why can we not do this?
Should we also not do this if the individual limits are finite values?
Is this specific to the sum law?

One of the reasons is that every time you have more and more terms/summands in the numerator which you can't control. In such cases we use either the squeeze theorem or, e. g., Stolz-Cesaro. — Matcha Latte, Jun 18 '20 at 14:51
$$\frac{1+1+\cdots+1}n$$ with $n$ ones on the top illustrates the same point in an even simpler way. — Angina Seng, Jun 18 '20 at 14:53

score 4 · Accepted Answer · answered Jun 18 '20 at 15:03

4

The sum law, or any of the other limit laws, is usually stated only for the case of 2 terms. By induction, we may extend it to 3, 4, 5, or any other fixed finite number of terms, where fixed means that the number does not depend on the limit variable $n$. However, as your example shows, we may not extend it to the case where the number of terms depends on $n$.

answered Jun 18 '20 at 15:03

Ted

35,732

Yes, we can. We can also extend it to any fixed, finite combination of sums, products, etc. – Ted Jun 18 '20 at 15:47

Why not to calculate the limit of individual terms when the number of terms approach infinity due to limiting variable?

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