Trivial Matters

My Calculus homework over the weekend was not very hard, so I decided to take it a bit further. We’re studying how to use power series to solve differential equations, and the instructor assigned us two problems. He said that for both we needed only to show the expanded form for y(x), and not try to find the pattern inherent to make it a power series again.

Not content with stopping there, I tried to analyze the patterns in each problem to find an appropriate power series representation. I spent all of my shift yesterday with my engineering pad, staring at the pattern 1, 4, 7, 10, …, trying to find a factorial representation for multiplying each term successively.

I tried to google the pattern, but all I could find was the obvious 3n – 2 notation, but no factorial. Then I found the On-Line Encyclopedia of Integer Sequences. I have a feeling that I’m going to be needing this site quite a bit as I progress in my program.

When I entered the products 1, 4, 28, 280, I found that the above notation was close, but needed a “triple factorial” to make it work. So the answer I was looking for is (3n – 2)!!! (that’s not emphasis, the notation requires three exclamation points). Our instructor had never mentioned the triple (or the double, for that matter) factorial, so I did a bit more research before using it. Also, I found that MAPLE only recognizes the double factorial, so I wasn’t able to verify my first answer.

Once I had that bit of information, the rest of the patterns fell into place. So I’ll probably be the only one in class tonight who has a power series representation for both answers. It would be nice if he’d just give me the A for the class (not that I won’t get one anyway), but at least I’d have one less thing to worry about.

Ok, gloating over now. You can return to your normal duties.

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