Question

Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a. (Enter your answers as a comma-separated list.)

f(x) = 7/(1+x), a = 2
Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.]
f(x) = e−5x
f(x)=
[infinity]
n = 0
=
Find the associated radius of convergence R.
R =

Answer 1

A) [ 7/3, (-7/9)(x/2), 7/27(x-2)^2, (-7/81)(x-2)^3 ]

B) attached below

Explanation:

A) Using the definition of a Taylor series

The first four nonzero terms of the series for f(x) = 7/ (1 +x), a = 2

= [ 7/3, (-7/9)(x/2), 7/27(x-2)^2, (-7/81)(x-2)^3 ]

attached below is the detailed solution

B) Finding Maclaurin series for f(x)

f(x) = e^-5x

attached below

Associated radius of convergence = ∞ ( infinity )