# Find the simplest counterexample against exchanging limit and summation

I would need a very simple counterexample to show that $$\lim_{M\to\infty}\sum_{t=1}^M f(t,M)$$ may not necessarily be equal to $$\sum_{t=1}^\infty \lim_{M\to\infty}f(t,M)\ .$$ The situation here is (slightly) different from the commonly asked question about interchanging limits and infinite summation, as $M$ is itself driving the upper limit of the sum. Can you exhibit a simple function $f$ which does the job? [Note that it should depend explicitly on $M$!]. I could only come up with an overly complicated situation, but I think I am missing something potentially very simple... Many thanks for you help.

• Are you looking for an example for which both expressions are finite? – Servaes Jan 10 '18 at 17:31
• @Servaes that would be ideal, yes! – Pierpaolo Vivo Jan 10 '18 at 17:32
• This works by using the standard sum $\sum_{k=1}^{n}1/n=1$. Just take $f(x, y) =1/y$ – Paramanand Singh Jan 11 '18 at 6:22
• It's not the simplest, but one important example is the Eisenstein series $G_{2k}$. (It's dealing with exchanging two sums rather than a sum and a limit, but the principle is the same). For $2k \geq 4$, the sum invovled converges uniformly, and the sums can be freely exchanged. For $2k = 2$ it fails, and extra term pops out that ruins the behavior under the modular group. (I mention it because a lot of students think that issues like this are pedantry or simply methods that are correct but verboten because they haven't been covered in class.) – anomaly Jan 11 '18 at 19:30

Using Iverson brackets, $$f(k,M)=[k=M]$$ $$%f(k,M)=\left\{\begin{array}{} %0&\text{if }k\ne M\\ %1&\text{if }k=M %\end{array}\right.$$

• Nice. Are the limits 1 and 0? – Eric Duminil Jan 11 '18 at 1:23
• @EricDuminil: indeed, the limits are $1$ and $0$. – robjohn Jan 11 '18 at 4:36
• In fact, you can get any two limits $a$ and $b$ you want by this method: $f(k, M) = 0$ for $M < k$, $a$ for $M = k$, and $b$ for $M > k$. – Michael Seifert Jan 11 '18 at 15:30
• @MichaelSeifert: the sum of the limits is infinite for that unless $b=0$, is it not? – robjohn Jan 11 '18 at 15:37
• @MichaelSeifert: $f(k,M)=a2^{-k}+(b-a)[k=M]$ or something like that would work. – robjohn Jan 11 '18 at 16:45

Let $f(t,M) =\frac{t}{M}$, then

$\lim_{M\rightarrow \infty} \sum_{t=1}^{M} \frac{t}{M} = \infty$

as it is just the arithmetic series over M,

$\lim_{M \rightarrow \infty} \frac{M(M+1)}{2M} = \lim_{M\rightarrow \infty} \frac{M+1}{2}$

while

$\sum^\infty_{t=1} \lim_{M\rightarrow \infty} \frac{t}{M} = 0$,

as every summand is zero for every finite $t$.

• I so far like this one the most. It's easy, well-known and natural to people new in calculus. – yo' Jan 11 '18 at 22:40
• How is $\lim_{M→+∞}∑_{t=1}^{M} \frac{t}{M}$ the harmonic series ? It is just $\lim_{M→+∞}\frac{M+1}{2}$. – Evpok Jan 12 '18 at 8:59
• You're right, I made a mistake on that one, it's the arithmetic series over M, i.e., $\frac{M(M+1)}{2M}$ Thanks! – cgrudz Jan 12 '18 at 13:04

One example I can think of is: $f(t,M)=\frac1M g(\frac tM)$ where $g$ is continuous on $[0,1]$, so that $$\lim_{M\rightarrow \infty}\sum_{t=0}^{M}f(t,M)=\int_0^1 g(x)\,\mathrm{d}x$$ but $\sum_{t=0}^{\infty}\lim_{M\rightarrow \infty}f(t,M)=0$ if $g(0)\neq 0.$
For example you can choose $g(x)=x+1\text{ and }f(t,M)=\frac1M (1+\frac tM).$
EDIT: Of course that's a Riemann series, I thought I'd mention it.

• It does! I just did a typo, sorry about that ! – deque Jan 10 '18 at 17:37

Similar to @cgrudz's answer but a bit simpler. $$f(t,M)=1/M$$ Then $\lim_{M\to\infty}\sum_{t=1}^M1/M=1$ while $\sum_{t=1}^\infty\lim_{M\to\infty}1/M=0$.