## Example 8.3.6 (b): Power Series Examples

Write the following series in sigma-notation and list the general term

*a*:_{n}-
*1 + 2x + 3x*^{2}+ 4x^{3}+ ... -
*1 - 1/2 x + 1/4 x*^{2}- 1/8 x^{3}+ 1/16 x^{4}... -
*3/2 x + 4/6 x*^{2}+ 5/24 x^{3}+ 6/120 x^{4}+ ...

Here we need to determine the general term *a _{n}* so
that we can rewrite the series in sigma notation as:

a_{n}(x - c)^{n}= a_{0}+ a_{1}(x-c) + a_{2}(x-c)^{2}+ ...

- For
*1 + 2x + 3x*we have:^{2}+ 4x^{3}+ ... *a*,_{0}= 1

*a*,_{1}= 2

*a*,_{2}= 3

*a*, ..._{3}= 4

so that*a*and_{n}= (n+1)*1 + 2x + 3x*^{2}+ 4x^{3}+ ... = (n+1) x^{n}- For
*1 - 1/2 x + 1/4 x*we have:^{2}- 1/8 x^{3}+ 1/16 x^{4}... *a*,_{0}= 1

*a*,_{1}= -1/2 = -1/2^{1}

*a*,_{2}= 1/4 = 1/2^{2}

*a*, ..._{3}= -1/8 = -1/2^{3}

so that*a*and_{n}= (-1)^{n}1/2^{n}*1 - 1/2 x + 1/4 x*^{2}- 1/8 x^{3}+ 1/16 x^{4}... = (-1)^{n}1/2^{n}x^{n}- For
*3/2 x + 4/6 x*we have:^{2}+ 5/24 x^{3}+ 6/120 x^{4}+ ... *a*(careful),_{0}= 0

*a*,_{1}= 3/2

*a*,_{2}= 4/6 = 4/3!

*a*,_{3}= 5/24 = 5/4!

*a*, ..._{4}= 6/120 = 6/5!

so that*a*and_{n}= (n+2)/(n+1)!

Note that the summation this time starts at*3/2 x + 4/6 x*^{2}+ 5/24 x^{3}+ 6/120 x^{4}+ ... = (n+2)/(n+1)! x^{n}*n=1*

For each series *c = 0*.