# calculus

Use division of power series to find the first three terms of the Maclaurin
series for y = sec x.

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1. 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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