Question

Find the remainder term Rₙ(x) for the Taylor series centered at 0 for the following functions. Find an upper bound for the magnitude of the remainder on the given interval for the given value of n. (The bound is not unique.)

ƒ(x) = ln (1 - x); bound R₃(x), for |x| < 1/2

Accepted Answer

Identify the function and the point about which the Taylor series is centered. Here, the function is $$f(x) = \ln(1 - x)$$ and the series is centered at $$0 ($$Maclaurin series). Recall the general form of the remainder term (Lagrange form) for the Taylor series of order $$n$$ centered at $$0$$:

$$R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1}$$

where $$c is $$some value between $$0$$ and $$x. $$Compute the derivatives of $$f(x) up to $$order $$n+1 = 4. $$For $$f(x) = \ln(1 - x)$$, the derivatives follow a pattern:

- $$f'(x) = -\frac{1}{1-x}$$
- $$f''(x) = -\frac{1}{(1-x)^2$}$$
- $$f^{(3)}(x$$) = -\frac{2}{$$(1-x)^3$$}$$
- f$$^{(4)}$$(x) = -\frac{6}{(1-x)^4$}$$

Use this to write $$f^{(4)}(c$$)$$ explicitly. Substitute f$$^{(4)}$$(c)$$ into the remainder formula:

$$R_3(x) = \frac{f^{(4)}(c)}{4!} x^4 = \frac{-6}{4! (1 - c)^4} x^4$$

Simplify the factorial and constants. To find an upper bound for $$|R_3$(x)| on $$the interval $$|x| \u003c \frac{1}{2}$$, note that $$c$$ lies between $$0$$ and $$x$$, so $$|c| \u003c \frac{1}{2}. $$Use this to find the maximum value of $$\frac{1}{|1 - c|^4} on $$this interval, then multiply by $$\frac{|x|^4}{4!}$$ and the absolute value of the constant to get the bound.

Find the remainder term Rₙ(x) for the Taylor series centered at 0 for the following functions. Find an upper bound for the magnitude of the remainder on the given interval for the given value of n. (The bound is not unique.)

ƒ(x) = ln (1 - x); bound R₃(x), for |x| < 1/2

Verified video answer for a similar problem:

Key Concepts

Taylor Series and Remainder Term

Lagrange Form of the Remainder

Bounding the Remainder on an Interval

Find the remainder term Rₙ(x) for the Taylor series centered at 0 for the following functions. Find an upper bound for the magnitude of the remainder on the given interval for the given value of n. (The bound is not unique.)ƒ(x) = ln (1 - x); bound R₃(x), for |x| < 1/2

Verified video answer for a similar problem:

Key Concepts

Taylor Series and Remainder Term

Lagrange Form of the Remainder

Bounding the Remainder on an Interval

Find the remainder term Rₙ(x) for the Taylor series centered at 0 for the following functions. Find an upper bound for the magnitude of the remainder on the given interval for the given value of n. (The bound is not unique.)

ƒ(x) = ln (1 - x); bound R₃(x), for |x| < 1/2