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# Series to functions Find the function represented by the following series, and find the interval of convergence of the series. (Not all these series are power series.) sum_{k=1}^inftyfrac{x^{2k}}{4^k}

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Series
asked 2020-11-10
Series to functions Find the function represented by the following series, and find the interval of convergence of the series. (Not all these series are power series.)
$$\sum_{k=1}^\infty\frac{x^{2k}}{4^k}$$

## Answers (1)

2020-11-11
We know that $$\sum_{k=0}^\infty x^k=\frac{1}{1-x}$$
Given function is $$\sum_{k=0}^\infty\frac{x^{2k}}{4^k}$$
It can be written as $$\sum_{k=0}^\infty\frac{x^{2k}}{4^k}=\sum_{k=0}^\infty(\frac{x^2}{4})^k$$
Comparing this with equation, we may write
$$\sum_{k=0}^\infty(\frac{x^2}{4})^k=\frac{1}{1-\frac{x^2}{4}}$$
$$=\frac{4}{4-x^2}$$ Now, for the function to be convergent,
$$|\frac{x^2}{4}|<1$$</span>
$$\Rightarrow|x^2|<4$$</span>
So the solution is $$-2 Therefore, the given series can be represented as \(\frac{4}{4-x^2}$$ and the interval of convergence is $$(-2,2)$$

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