How do I find the sum of a Taylor series sum_(n=0)^oo((-1)^n(x-2)^n)/(2^(n+1))?

How do I find the value of $f \left(x\right) = {\sum}_{n = 0}^{\infty} \frac{{\left(- 1\right)}^{n} {\left(x - 2\right)}^{n}}{{2}^{n + 1}}$. The interval of convergence is (0,4)

Apr 10, 2018

See below.

Explanation:

For $\left\mid y \right\mid < 1$ we have

${\sum}_{k = 0}^{\infty} {\left(- 1\right)}^{k} {y}^{k} = \frac{1}{1 + y}$

considering now

$y = \frac{x - 2}{2}$

we have

${\sum}_{n = 0}^{\infty} \frac{{\left(- 1\right)}^{n} {\left(x - 2\right)}^{n}}{{2}^{n + 1}} = \frac{1}{2} \left(\frac{1}{1 + \frac{x - 2}{2}}\right)$

for $\frac{\left\mid x - 2 \right\mid}{2} < 1$