Question

Identify the functions represented by the following power series. $$\sum_{k=0}^{\infty} 2^{k} x^{2 k+1}$$

Short answer

Answer: The function represented by the given power series is \(g(x) = \frac{1}{(1-x)^{-\frac{1}{2}}}\).

Step-by-step solution

Rewrite the series using a common power

Since the power series has a term x^(2k+1), we can replace (2k+1) with a common power 'n' to make the series look like a common Maclaurin series representation. $$\sum_{k=0}^{\infty} 2^{k} x^{2 k+1} = \sum_{n=1}^{\infty} a_n x^{n}$$ with n = 2k+1 and hence k = (n-1)/2.

Identify the general term for the given power series

From the above equation and replacing k = (n-1)/2 we get, $$a_n = 2^{(n-1)/2}$$

Recognize the Maclaurin series of the known function

We need to recognize the above power series as the Maclaurin series (Taylor series around x=0) of a known function. The Maclaurin series for the function g(x) is given by: $$g(x) =\sum_{n=0}^{\infty} \frac{g^{(n)}(0)}{n!}x^n$$ where g^(n)(0) is the n-th derivative of g(x) evaluated at x = 0. Our goal is to find such a function g(x) using the coefficients we've found before.

Compare the series terms with known functions and find the target function.

We can compare our power series with the known Maclaurin series of the function \(\frac{1}{(1-x)^{\alpha}}\), which is given as: $$\frac{1}{(1-x)^{\alpha}} = \sum_{n=0}^{\infty} \binom{\alpha + n - 1}{n} x^n$$ For our problem, we can simplify the known function's power series to match our power series $$\sum_{n=1}^{\infty} \binom{\alpha + n - 1}{n} x^n = \sum_{n=1}^{\infty} a_n x^{n}$$ $$\binom{\alpha + n - 1}{n} = 2^{(n-1)/2}$$ Observing the coefficients, we notice that for odd values of n (which is 2k+1), the binomial coefficients become zero for the even powers of x (which don't exist in this power series). To achieve this, we want to choose the parameter α such that the power series only contains odd powers of x. By choosing α = -1/2, it turns the binomial coefficient into simple 2^(n-1)/2. Therefore, the function that we are looking for is: $$g(x) = \frac{1}{(1-x)^{-\frac{1}{2}}}$$

Find the function represented by the given power series

Finally, we found that our given power series is the Maclaurin series representation of the function given by: $$g(x) = \frac{1}{(1-x)^{-\frac{1}{2}}}$$