Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] f(x) = 4 cos(x), a = 7π

In this exercise we must calculate the Taylor series for the given function in this way;

The Taylor series of f(x) around the point a, can be written as:

Here we have:

Then, let's calculate each part:

Here we already can see two things:

1) The odd derivatives will have a sin(x) function that is zero when evaluated in .

2) We also can see that the sign will alternate between consecutive terms.

So we only will work with the even powers of the series:

So we can write it as:

Such that the n-th term can written as:

See more abour Taylor series at: link

The Taylor series of f(x) around the point a, can be written as:

Here we have:

f(x) = 4*cos(x)

a = 7*pi

then, let's calculate each part:

f(a) = 4*cos(7*pi) = -4

df/dx = -4*sin(x)

(df/dx)(a) = -4*sin(7*pi) = 0

(d^2f)/(dx^2) = -4*cos(x)

(d^2f)/(dx^2)(a) = -4*cos(7*pi) = 4

Here we already can see two things:

the odd derivatives will have a sin(x) function that is zero when evaluated in x = 7*pi, and we also can see that the sign will alternate between consecutive terms.

so we only will work with the even powers of the series:

f(x) = -4 + (1/2!)*4*(x - 7*pi)^2 - (1/4!)*4*(x - 7*pi)^4 + ....

So we can write it as:

f(x) = ∑fₙ

Such that the n-th term can written as:

x=half

step-by-step explanation: