Find the power series for $f(x) = \frac{1}{1+4x^6}$. $\sum_{n=0}^{\infty}$ Enter your answer $\sqrt{x}$ Identify its interval of convergence. The series is convergent from $x=$ Enter your answer $\sqrt{x}$ to $x=$ Enter your answer $\sqrt{x}$ (Round your answer to two decimal places)
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Step 1: We can write the given function as a geometric series: $f(x) = \frac{1}{1+4x^6} = \frac{1}{1-(-4x^6)} = \sum_{n=0}^{\infty} (-4x^6)^n = \sum_{n=0}^{\infty} (-1)^n 4^n x^{6n}$. Show more…
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