Question

Find a geometric power series for the function, centered at 0 , (a) by the technique shown in Examples 1 and 2 and (b) by long division. $$ f(x)=\frac{1}{2-x} $$

Short answer

The geometric power series representation for the function \( f(x) = \frac{1}{2-x} \) is \( \sum_{n=0}^\infty (-1)^n (x-2)^n \), valid for \( |x-2|<1 \).

Step-by-step solution

Express the function in the form of a geometric series

To start with, express the function \( f(x) \) to match the shape of the formula for the sum of infinite geometric series. The common ratio 'r' of the geometric series in this case is x, and 'a' is 1/(2-x). Therefore, we can rewrite \( f(x) \) in the standard form of a geometric series sum formula \[ f(x) = \frac{1}{2-x} = \frac{1}{1- (-x+2)} = \frac{1}{1-(-(x-2))} \]

Apply the geometric series sum formula

Now, applying the geometric series sum formula, we get \[ f(x) = \sum_{n=0}^\infty (-(x-2))^n \], provided |-(x-2)| < 1.

Simplify the expression

The series can further be simplified by removing the negative sign as it can alternate the sign of each term in the series. Hence, \[ f(x) = \sum_{n=0}^\infty (-1)^n (x-2)^n \].

Perform long division

Use the method of polynomial long division, for \( f(x) = \frac{1}{2-x} \), treat 1 as a polynomial and '2-x' as another one. If we divide these polynomials, we get the power series representation of f(x), which will match the one obtained in the previous steps.

Write the final answer

Finally, we know from the steps above, the geometric series representation for \( f(x) \) is \[ f(x) = \sum_{n=0}^\infty (-1)^n (x-2)^n \].