## Example 8.3.6 (a): Power Series Examples

*a*and

_{3}*a*for each:

_{4}- (-1)
^{n}(x+2)^{n} - (x+2)
^{2n} - (2x+2)
^{2n}

A power series looks as follows:

a_{n}(x - c)^{n}= a_{0}+ a_{1}(x-c) + a_{2}(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,a, and_{3}= -1a._{4}= 1

The second series is:

(x+2)^{2n}= 1 + (x+2)^{2}+ (x+2)^{4}+ ...But since

ais the coefficient in front of the_{n}n-th power, we need to include the 'missing' coefficients as zeros. Thus:c = -2,a, and_{3}= 0a._{4}= 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

xto stand alone. Thus we need to re-write the series:

(2x+2)^{2n}= (2(x+1))^{2n}= 2^{2n}(x+1)^{2n}= 1 + 2^{2}(x+1)^{2}+ 2^{4}(x+1)^{4}+ ...Now we can determine the coefficients:

c = -1,a, and_{3}= 0a._{4}= 2^{4}