Question

In each of Problems 7 through 13, determine the Taylor series about the point xo for the given function. Also determine the radius of convergence of the series. 7. sinx, xo =0 8. et, Xo = 0 9. x, xo = 1 10. x2, xy = -1 11. Inx, x0 = 1 12. x0=0

Answer 1

The Taylor series for
`\( \sin(x) \)` about
`\( x_0 = 0 \)` is
`\( \sum_(n=0)^(\infty) ((-1)^n x^(2n+1))/((2n+1)!) \), with \( R = \infty \).` The Taylor series for
`\( \ln(x) \)` about
`\( x_0 = 1 \)` is
`\( \sum_(n=1)^(\infty) ((-1)^(n+1)(x-1)^n)/(n) \), with \( R = 1 \).`

Step-by-step explanation:

To determine the Taylor series for each function, we use the formula
`\( f(x) = \sum_(n=0)^(\infty) (f^((n))(x_0))/(n!)(x - x_0)^n \)` , where
`\( f^((n))(x_0) \)` denotes the
`\(n\)-th` derivative of f(x) evaluated at
`x_0` . Calculating the derivatives and applying the formula yields the given Taylor series for each function.

The radius of convergence R is determined by the ratio test,
`\( \lim_(n \to \infty) (\left|a_(n+1)\right|)/(\left|a_n\right|) \), where \( a_n \)` is the coefficient of
`\( (x - x_0)^n \)` in the Taylor series. For the functions given, the ratio test results in
`\( R = \infty \)` for all except
`\( \ln(x) \) at \( x_0 = 1 \)`, where R = 1 .

Understanding the Taylor series and radius of convergence is fundamental in representing functions as power series and determining the range of values for which the series converges.

Answer 2

The question seeks the Taylor series representations and their radii of convergence for various functions at specific points. For instance, the Taylor series for sin(x) at x0=0 has an infinite radius of convergence.

Step-by-step explanation:

The provided question asks for the determination of the Taylor series expansion of various functions about specified points x0, and also the computation of the radius of convergence for these series. For example, let's discuss the Taylor series for sin(x) about x0 = 0. The Taylor series for a function f(x) around a point a is given by:

• f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + f'''(a)(x-a)³/3! + ...

Using this formula, the Taylor series for sin(x) about 0 is:

• sin(x) = x - x³/3! + x⁵/5! - x⁷/7! + ...

This series is the sum of all derivatives of sin(x) evaluated at 0, with each term having alternating signs and being divided by the factorial of its corresponding odd integer. The radius of convergence for this series is infinite, meaning it converges for all real numbers.