# How a visualization can make all the difference

Here comes a flash post from the airport in Zürich. Ready? Go!

It’s a well-known fact that different visualizations of the same data often give varied information about the data and it’s properties. It’s also a well-known fact that when the visualization is not natural with respect to the data, you may not even get any information at all.

I was looking for an elementary example to show that, when I checked out the college mathematics journal (of MAA). In it, there was a problem with a statement something along the lines,

Study the properties of the

$\lim_{n\to\infty}S_n=\lim_{n\to\infty}\sum_{k=1}^{n}sin(k)$

Can you give upper/lower bounds to the sum? Does the sum even converge?

So, if we plot the first 1000 elements of the sequence $S_k$, we get

Which gives us an idea of the bounds of the sum but tells us nothing of the distribution of the elements of the sequence. Then again, this sequence is a discrete structure and in this case, it may seem kind of unnatural to use a line to visualize the sequence.  By going for points instead, we get,

and wow! We immediately get a useful pattern. We can now study how the circles meet, if we can overlap them with multiples of the sine function and generally work with that to get more results. 🙂

PS : The plots were done with PyLab. I’ve been trying to balance my use of matlab with that of python.