So, if I ever make a list with the ten most counter-intuitive things (at least, at first glance), remind me to put into number 4 this limit :
I won’t lie. The first time that I looked at this beast, a small voice inside my head was saying “does not exist”. The reason was that because of the sine and it’s periodicity, I was waiting for the frequency with which it oscillates to simply screw things up in the same way that the limit
I was also aware of limits like but I didn’t make the connection at the time.
So you can imagine how I was when I found out that this limit does exist and is equal to 2π. Since I didn’t have a photographer near, this image will suffice.
I will give you links with the proof at the end. For me, the important part is to have first a good intuition on why this works like that.
So, let’s forget at this point about 2π and focus on the part. Let’s also keep in mind that and that we can break that sum into two parts. By multiplying both by n!, we get a sum for the integer part and one for the fractional part of .
Let’s see what happens!
Supposing that is the sequence of integer parts of and is the sequence of fractional parts, we get
Thus, we just need to find the limit of and, hopefully, we will be done. What should be the limit of ?
Returning to the series, we can write that fractional part as
from which we can see nice and well that while (try to bound it).
Thus, and, by using the idea mentioned earlier about , we can work our way to the full result.