Question

c) Find the Mclaurin series for the function
ELIF
up to 5 terms​

Answer 1

Answer:

Maclaurin Series

A special case arises when we take the Taylor series at the point 0. When we do this, we get the Maclaurin series. The Maclaurin series is the Taylor series at the point 0. The formula for the Maclaurin series then is this:

maclaurin series

We use this formula in the same way as we do the Taylor series formula. We find the derivatives of the original function, and we use those derivatives in our series when it calls for it. The only difference is that we are now strictly using the point 0. All our derivatives are evaluated at the point 0.

ex

What is the Maclaurin series for the function f(x) = ex?

To find the Maclaurin series for this function, we first find the various derivatives of this function. This particular function is actually a very interesting function. All of its derivatives in fact are itself. So, the first derivative is ex, the second derivative is ex, and so on. Since we are looking at the Maclaurin series, we need to evaluate this function ex at the point 0. Since all the derivatives are the same, we evaluate ex at x = 0. We get e0 = 1. So all our derivatives will equal 1. Our Maclaurin series then becomes this:  

maclaurin series

What we did was plug in 1 for all the derivatives since all our derivatives evaluated at the point 0 is equal to 1. The last two lines are our answer. The last line is the series written in summation form, and the line before that is the series expanded.

sin x

Find the Maclaurin series for f(x) = sin x:

To find the Maclaurin series for this function, we start the same way. We find the various derivatives of this function and then evaluate them at the point 0. We get these for our derivatives:

Derivative At the point 0

f(x) = sin x f(0) = 0

f'(x) = cos x f'(0) = 1

f''(x) = -sin x f''(0) = 0

f'''(x) = -cos x f'''(0) = -1

f''''(x) = sin x f''''(0) = 0

We see our derivatives following a pattern of 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, etc. The numbers 0, 1, 0, and -1 keep repeating.

our last two lines are the answer with the last line being our answer written in summation form, and the line before that being our series expanded. Since we have the zeroes, when we write our answer, we skip over the zeroes...

Step-by-step explanation:

tnk u  :)