Question

Any methoda. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients.b. Determine the radius of convergence of the series.f(x) = (1 + x²)⁻²/³

Accepted Answer

Identify the function given: $$f(x) = (1 + x$$^{2}$$)^{-\frac{2}$${3}}$$. We want to find the first four nonzero terms of its Taylor series centered at 0 (Maclaurin series). Recall the generalized binomial series expansion for (1$$ + $$u)^{k}$$ where $$k is $$any real number:

$$
(1 + u)^{k} = \sum_{n=0}^{\infty} \binom{k}{n} u^{n} = 1 + k u + \frac{k(k-1)}{2!} u^{2} + \frac{k(k-1)(k-2)}{3!} u^{3} + \cdots
$$

Here, $$u = x$$^{2}$$ and k$$ = -\frac{2}{3}$$. Write out the first four terms explicitly by substituting u$$ = $$x^{2}$$ and calculating the binomial coefficients:

- Term 0: $$\binom{k}{0} u$$^{0}$$ = 1
- $$Term 1: $$\binom{k}{1} u$$^{1}$$ = k x$$^{2}$$
- Term 2: $$\binom{k}{2} $$u^{2}$$ = \frac{k(k-1)}{2!} $$x^{4}
- $$Term 3: $$\binom{k}{3} u$$^{3}$$ = \frac{k(k-1)(k-2)}{3!} x$$^{6}$$ Substitute k$$ = -\frac{2}{3}$$ into each coefficient and simplify the expressions for the coefficients of x$$^{2}$$, x$$^{4}$$, and x$$^{6}$$. This will give the first four nonzero terms of the Taylor series:

f(x) = 1$$ + \left(-\frac{2}{3}\right) x^{2} + \frac{\left(-\frac{2}{3}\right)\left(-\frac{5}{3}\right)}{2} x^{4} + \frac{\left(-\frac{2}{3}\right)\left(-\frac{5}{3}\right)\left(-\frac{8}{3}\right)}{6} x^{6} + \cdots
$$ To find the radius of convergence, note that the binomial series for (1$$ + $$u)^{k}$$ converges when $$|u| \u003c 1. $$Since $$u = x$$^{2}$$, the radius of convergence in terms of x$ is determined by $$|$$x^{2}$$| \u003c 1$$, which simplifies to $$|x| \u003c 1$. Thus, the radius of convergence is 1.

Any method

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients.

b. Determine the radius of convergence of the series.

f(x) = (1 + x²)⁻²/³

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Concetti chiave

Taylor Series Expansion

Binomial Series for Non-Integer Exponents

Radius of Convergence