# 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} Question
Series 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.)
$$\sum_{k=1}^\infty\frac{x^{2k}}{4^k}$$ 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)$$

### Relevant Questions 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.)
$$\sum_{k=0}^\infty(\frac{x^2-1}{3})^k$$ Representing functions by power series Identify the functions represented by the following power series.
$$\sum_{k=1}^\infty\frac{x^{2k}}{k}$$ Power series for derivatives
a. Differentiate the Taylor series centered at 0 for the following functions.
b. Identify the function represented by the differentiated series.
c. Give the interval of convergence of the power series for the derivative.
$$f(x)=\tan^{-1}x$$ Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
$$\sum_{k=0}^\infty\frac{(x-1)^k}{k!}$$ Find the interval of convergence of the power series, where c > 0 and k is a positive integer.
$$\sum_{n=1}^\infty\frac{n!(x-c)^n}{1\cdot3\cdot5\cdot...\cdot(2n-1)}$$ Find the interval of convergence of the power series, where c > 0 and k is a positive integer.
$$\displaystyle{\sum_{{{n}={1}}}^{\infty}}{\frac{{{n}!{\left({x}-{c}\right)}^{{n}}}}{{{1}\cdot{3}\cdot{5}\cdot\ldots\cdot{\left({2}{n}-{1}\right)}}}}$$ Identify a convergence test for the following series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.
$$\sum_{k=3}^\infty\frac{2k^2}{k^2-k-2}$$ Power series for derivatives
a. Differentiate the Taylor series centered at 0 for the following functions.
b. Identify the function represented by the differentiated series.
c. Give the interval of convergence of the power series for the derivative.
$$f(x)=\ln(1+x)$$ a.Find the first four partial sums $$S_1,S_2,S_3,$$ and $$S_4$$ of the series.
b.Find a formula for the nth partial sum $$S_n$$ of the indinite series.Use this formula to find the next four partial sums $$S_5,S_6,S_7$$ and $$S_8$$ of the infinite series.
$$\sum_{k=1}^\infty\frac{2}{(2k-1)(2k+1)}$$ $$\sum_{n=0}^\infty\frac{2^n(x-3)^n}{\sqrt{n+3}}$$