Representing Functions as Power Series (Notes #19 - Section 11.9)

[@SerpentineIntegral]

There is a mistake at 50:14. (x/5) should carry over into the series expression as (-x/5) because the geometric sum formula is a / (1 - r). This changes the final series for f(x) into SIGMA (-1)^(n+1)*n*x^(n-1) / 5^(n+1). The radius of convergence is not affected by this mistake, though. Stupid sign error!

[@mf_01]

At 50:14 should it be (-1/5) / (1 - (-x/5)) because it should match a / (1 - r)? And then that would change the sign of the answer series?

[@SerpentineIntegral]

This is why I miss the live classes... Yes! You are correct; the final series should be f(x) = sigma (-1)^(n+1)*n*x^(n-1) / 5^(n+1). My mistake.

[@jessicafrederick2524]

At 1:17:57, how do you know whether it should be n=0 or it should n=1 like it was at 1:01:08?

[@SerpentineIntegral]

We often switch from n=0 to n=1 after taking the derivative of a power series because taking a derivative makes the "zeroth" term (which is a constant) become equal to 0. But at 1:17:57, we took an integral, which, unlike a derivative, does not "zero out" the constant term. In fact, taking an integral doesn't make ANY terms equal 0, so we can't simply drop an index in that situation. The general principle at work is if the n=0 thru n=k terms of a series are just zero, you can skip those terms and just start the series at n=k+1. However, dropping the n=0 term after taking a derivative is _mostly_ just a polite courtesy. You can leave the series starting at n=0 if you like, it just means you (or the computer) are going to end up adding some initial zero terms you don't need to. Actually, there is one other reason to drop the initial term, but really it's just a technicality: if you start the index at n=0 after taking a derivative, and then you plug 0 in for x, the n=0 term of the series contains a factor of 0^(-1) which is undefined. However, generally the whole point of power series is to plug in values of x OTHER THAN the center point of the series, which is why this is really not a big deal. Bottom line: Dropping the n=0 term after taking a derivative is optional (almost always).