Examples where we cannot interchange summation and integration

Question

We know of conditions under which we can interchange summation and integration (1, 2).

What are some simple examples where we cannot do so and which we could present to high-school/introductory calculus students (so as to warn them against always simply assuming that we can do so)?

Answer 1

Let $f_n(x)$ be the triangle function of height $1$ and width $2$, centred on $x=n$. This is what $f_6$ looks like:

Then

• $\int_{-\infty}^\infty f_n(x)\,dx=1$ for all $n$
• $\lim_{n\rightarrow\infty}f_n(x)=0$ for all $x$

So $$0=\int_{-\infty}^\infty \lim_{n\rightarrow\infty}f_n(x)\,dx\ne \lim_{n\rightarrow\infty}\int_{-\infty}^\infty f_n(x)\,dx=1$$

Edited to add: I see now that you asked for a series, not a sequence. So replace the sequence $(f_n)$ in the above with the series $\sum g_n$, defined by $g_1=f_1$ and $g_{n+1}=f_{n+1}-f_n$. Now $\sum_{r=1}^n g_r=f_n$.