# Interactive Real Analysis - part of MathCS.org

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## Example 8.3.6 (b): Power Series Examples

Write the following series in sigma-notation and list the general term an:
1. 1 + 2x + 3x2 + 4x3 + ...
2. 1 - 1/2 x + 1/4 x2 - 1/8 x3 + 1/16 x4 ...
3. 3/2 x + 4/6 x2 + 5/24 x3 + 6/120 x4 + ...

Here we need to determine the general term an so that we can rewrite the series in sigma notation as:

an (x - c)n = a0 + a1(x-c) + a2(x-c)2 + ...
For 1 + 2x + 3x2 + 4x3 + ... we have:

a0 = 1,
a1 = 2,
a2 = 3,
a3 = 4, ...
so that an = (n+1) and

1 + 2x + 3x2 + 4x3 + ... = (n+1) xn

For 1 - 1/2 x + 1/4 x2 - 1/8 x3 + 1/16 x4 ... we have:

a0 = 1,
a1 = -1/2 = -1/21,
a2 = 1/4 = 1/22,
a3 = -1/8 = -1/23, ...
so that an = (-1)n 1/2n and

1 - 1/2 x + 1/4 x2 - 1/8 x3 + 1/16 x4 ... = (-1)n 1/2n xn

For 3/2 x + 4/6 x2 + 5/24 x3 + 6/120 x4 + ... we have:

a0 = 0 (careful),
a1 = 3/2,
a2 = 4/6 = 4/3!,
a3 = 5/24 = 5/4!,
a4 = 6/120 = 6/5!, ...
so that an = (n+2)/(n+1)! and

3/2 x + 4/6 x2 + 5/24 x3 + 6/120 x4 + ... = (n+2)/(n+1)! xn
Note that the summation this time starts at n=1

For each series c = 0.

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