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

That's ... helpful?

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,

Wow!

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.

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