# Interactive Real Analysis

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## 8.3. Series and Power Series

### Example 8.3.6 (a): Power Series Examples

All of the following series are power series. List the coefficients a3 and a4 for each:
1. (-1)n(x+2)n
2. (x+2)2n
3. (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.

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