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Rama09 [41]
3 years ago
11

Find a power series representation for the function. (Assume a > 0.) (Give your power series representation centered at x = 0

.) f(x) = x5 a6 − x6
Mathematics
1 answer:
wariber [46]3 years ago
8 0

Answer:

The answer is "a_n =(\frac{x^{8n+5}}{a^{8n+8}})"

Step-by-step explanation:

f(x) =\frac{x^5}{a^8 -x^8} = \frac{\frac{x^5}{a^8}}{1-(\frac{x}{a})^8}\\\\\\\to \frac{1}{1-(\frac{x}{a})^8}

The above equation can be writtern as the sum of the \infty geometric series with r= \frac{x}{a}

= \frac{\frac{x^5}{a^8}}{1+(\frac{x}{a})^8} \\\\ = \frac{x^5}{a^8} [ 1+ (\frac{x}{a})^8 + ((\frac{x}{a})^8)^2+.......]\\\\= \frac{x^5}{a^8}  \sum_{\infty}^{n=0}  (\frac{x}{a})^{8n}\\\\ = \sum_{\infty}^{n=0}  (\frac{x^{8n+5}}{a^{8n+8}})\\\\a_n =(\frac{x^{8n+5}}{a^{8n+8}})

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