Use division of power series to find the first three terms of the Maclaurin
series for y = sec x.
There are different ways of doing the division. A method closely related to long division is to write down the general form of the answer in terms of undetermined coefficients and then multiply both sides by the numerator and then solve for the coefficients by equating equal powers of x.
In this case, we can put:
1/cos(x) = a0 + a2 x^2 + a4 x^4 + a6 x^6+...
We know that the function is an even function, so we only have even powers of x. Multiply bot sides by cos(x):
1 = [a0 + a2 x^2 + a4 x^4 + a6 x^6+...]
[1 - x^2/2 + x^4/4! - x^6/6! + ...] =
a0 + (a2 - a0/2)x^2 +
(a4 - a2/2 + a0/4!) x^4 +
(a6 - a4/2 + a2/4! - a0/6!) x^6 + ..
The constant term has to be 1:
a0 = 1
The coefficient of x^2 has to be zero:
a2 - a0/2 = 0 -->
a2 = a0/2 = 1/2
The coefficient of x^4 has to be zero:
a4 - a2/2 + a0/4! = 0 -->
a4 = a2/2 - a0/4! = 1/4 - 1/4! = 5/24
The coefficient of x^6 has to be zero:
a6 - a4/2 + a2/4! - a0/6! = 0 --->
a6 = a4/2 - a2/4! + a0/6! = 61/720
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To find the Maclaurin series for y = sec x, we can start by expressing sec x in terms of its reciprocal function, cos x:
sec x = 1/cos x
Now, let's find the power series representation for cos x and then take the reciprocal to get the series for sec x.
Step 1: Finding the power series for cos x
The power series representation for cos x is given by:
cos x = 1 - x^2/2! + x^4/4! - x^6/6! + ...
Step 2: Taking the reciprocal to find the series for sec x
To convert the series for cos x to the series for sec x, we need to take the reciprocal of each term. Since the first term of the cos x series is 1 and we are taking the reciprocal, the first term of the sec x series will also be 1.
Reciprocals of the remaining terms:
The reciprocal of (-x^2/2!) is -2/1! * x^2 = -x^2
The reciprocal of (x^4/4!) is 4!/x^4 = 24/x^4
The reciprocal of (-x^6/6!) is -6!/x^6 = -720/x^6
Step 3: Write the series for sec x
Putting all the terms together, we get:
sec x = 1 - x^2 + (24/x^4) - (720/x^6) + ...
The first three terms of the Maclaurin series for y = sec x are:
1 - x^2 + 24/x^4
Note that the series expansion should be carried out to as many terms as needed for the desired level of accuracy.