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Igoryamba
2 years ago
5

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.)
Mathematics
1 answer:
babunello [35]2 years ago
4 0

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)

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