Video transcriptWe've already seen manyexamples of infinite series. But what's exciting about whatwe're about to do in this video is we're going to use infiniteseries to define a function. And the most commonone that you will see in your mathematicalcareers is the power series. And I'm about to write a generalcase of the power series. So I could imaginea function, f of x, being defined asthe infinite sum. So going from n equals 0to infinity of a sub n-- so a sub n is just going to bethe coefficient on each term-- times our variable xminus some constant c. You could almostimagine this is shifting our function to the n-th power. So if I were toexpand this out, I have my first term'scoefficient, a sub 0, times x minus cto the 0-th power, plus a sub 1 times x minusc to the first power. This one, of course, willsimplify just a sub 0. This would simplify to a sub1 times x minus c plus a sub 2 times x minus c squared. And I could just keepgoing on and on and on. Now, when you seethis, you mig...
