Example 8.3.12 (b): Differentiating and Integrating Power Series
The series converges for all x (confirm!) and can therefore be integrated (and differentiated) term-by-term. We have:
and integrating again:
Now, the above integration is technically correct, but with all the summation signs, integrals, and indeces it is difficult to see what is really going on. Things get a little clearer if we write out the series and integrate 'manually':
f(x) = =
= 1 - 1/2! x2 + 1/4! x4 - 1/6! x6 + ...
Integrating gives us a new function g(x):
g(x) = f(x) dx = x - 1/3! x3 + 1/5! x5 - 1/7! x7 + ... + C
Integrating again gives us:
g(x) dx = 1/2 x2 - 1/4! x4 + 1/6! x6 - 1/8! x8 + ... + Cx + D =
= -f(x) + 1 + Cx + D' =
= -f(x) + Cx + D''
where we subsume the +1 into the new constant D'.
In other words, if we ignore the constants we have two functions f and g such that
f(x) dx = g(x) and g(x) dx = -f(x)
Hmmm ... what well-known functions could f and g be? To help your memory, here is the graph of these functions:
f(x) = g(x) =