Question

Find a power series for the function, centered at c. 7x c=0 x2+5x 6 , g(x) - Determine the interval of convergence. (Enter your answer using interval notation.)

Answer 1

Answer:

$\mathbf{g(x) = \sum \limits^{\infty}_{n=0} (-1 + (\dfrac{-1}{6})^n)x^n }$

and the interval of the convergence is (-1, 1)

Step-by-step explanation:

To find a power series for the function, centered at c.

$g(x) = \dfrac{7x}{x^2 +5x-6} ,\ c = 0$

If we factorize the denominator, we have:

$g(x) = \dfrac{7x}{(x +6)(x-1)}$

$g(x)= \dfrac{1}{x-1}+\dfrac{6}{x+6}$

Thus;

$g(x)= \dfrac{-1}{1-x}+\dfrac{1}{1+\dfrac{x}{6}}$

$g(x)= \dfrac{-1}{1-x}+\dfrac{1}{1-(-\dfrac{x}{6})}$

$g(x) = - \sum \limits^{\infty}_{n=0} x^n + \sum \limits^{\infty}_{n=0} x^n(\dfrac{-x}{6})^n \ \ if \ \ |x| < 1 \ \ and \ \ |\dfrac{x}{6}< 1$

$g(x) = \sum \limits^{\infty}_{n=0} (-1 + (\dfrac{-1}{6})^n)x^n , \ if |x|$

$\mathbf{g(x) = \sum \limits^{\infty}_{n=0} (-1 + (\dfrac{-1}{6})^n)x^n }$

and the interval of the convergence is (-1, 1)