Example 8.3.6 (a): Power Series Examples
- (-1)n(x+2)n
- (x+2)2n
- (2x+2)2n
A power series looks as follows:
an (x - c)n = a0 + a1(x-c) + a2(x-c)2 + ...
The first series above is:
(-1)n(x+2)n = 1 - (x+2) + (x+2)2 - (x+2)3 + ...
Thus: c = -2, a3 = -1, and a4 = 1.
The second series is:
(x+2)2n = 1 + (x+2)2 + (x+2)4 + ...
But since an is the coefficient in front of the n-th power, we need to include the 'missing' coefficients as zeros. Thus: c = -2, a3 = 0, and a4 = 1.
The third series is:
(2x+2)2n = 1 + (2x+2)2 + (2x+2)4 + ...
Here we again have missing (i.e. zero) coefficients, but we also need our x to stand alone. Thus we need to re-write the series:
(2x+2)2n = (2(x+1))2n = 22n(x+1)2n = 1 + 22(x+1)2 + 24(x+1)4 + ...
Now we can determine the coefficients: c = -1, a3 = 0, and a4 = 24.