College Math Teaching

September 20, 2013

Ok, have fun and justify this…

Filed under: calculus, popular mathematics, Power Series, series, Taylor Series — Tags: — collegemathteaching @ 7:59 pm


Ok, you say, “this works”; this is a series representation for \pi . Ok, it is but why?

Now if you tell me: \int^1_0 \frac{dx}{1+x^2} = arctan(1) = \frac{\pi}{4} and that \frac{1}{1+x^2} =  \sum^{\infty}_{k=0} (-1)^k x^{2k} and term by term integration yields:
\sum^{\infty}_{k=0} (-1)^k \frac{1}{2k+1}x^{2k+1} I’d remind you of: “interval of absolute convergence” and remind you that the series for \frac{1}{1+x^2} does NOT converge at x = 1 and that one has to be in the open interval of convergence to justify term by term integration.

True, the series DOES converge to \frac{\pi}{4} but it is NOT that elementary to see. 🙂


(Yes, the series IS correct…but the justification is trickier than merely doing the “obvious”).


Leave a Comment »

No comments yet.

RSS feed for comments on this post. TrackBack URI

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Create a free website or blog at

%d bloggers like this: