Get the answer to "Determine the interval of convergence and find a..."

ossidianaZ

Answered question

2021-10-25

Determine the interval of convergence and find a power series representation for the function.
$f (x) = \frac{2}{3} - x$

Answer & Explanation

SkladanH

Skilled2021-10-26Added 80 answers

$f (x) = \frac{2}{3 - x}$
Divide Numerator and denominator by 3. To get
$f (x) = \frac{\frac{2}{3}}{1 - \frac{x}{3}}$
Keep in mind that the geometric series' sum with the initial term's common ratio and r is
$S = \sum_{n = 0}^{\infty} a r^{n} = \frac{a}{1 - r}$
The given function can be interpred as
$\frac{\frac{2}{3}}{1 - \frac{x}{3}} = \frac{a}{1 - r}$
Therefore, we can say that f(x) is a sum of a geometric series with initial term $a = \frac{2}{3}$ and common ratio $r = \frac{x}{3}$
Therefore,
$f (x) = \sum_{n = 0}^{\infty} a r^{n} = \sum_{n = 0}^{\infty} (\frac{2}{3}) {(\frac{x}{3})}^{n} = \sum_{n = 0}^{\infty} (\frac{2}{3^{n + 1}}) \cdot x^{n}$
This is the power series representation of f(x)
We know that the geometric series converges when $| r | = | \frac{x}{3} | < 1$
$| x | < 3$
Intterval of converges is $(- 3, 3)$
Radius of converges is 3

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