# calculus

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

1. 👍
2. 👎
3. 👁
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

1. 👍
2. 👎
2. need answer to

tinyurl slash x x 2 3 5 h j

1. 👍
2. 👎

## Similar Questions

1. ### Arithmetic Series - - Reiny

Consider the series defined by Sn = 3n-1 Find the first four terms of the series. How exactly do I do this?

2. ### Mathematics

The sum of the first n terms os the series is given Sn=2n(n+3). Show that the terms of the series forms an arithmetic progression.

3. ### mathematics

the 1st term of arithmetic series is 2 .the 1st ,3rd and 11th terms are the first the terms of a geometric series. find the 6 terms of the arithemetic series

4. ### mathematics

The common ratio of a geometric series is 1/3 and the sum of the first 5 terms is 121. a) What is the value of the first term? b) Write the first 5 terms of the series.

1. ### math

The sum of the 1st nine terms of an arithmetic series is 216. The 1st,3rd and the 7th terms of series form the 1st three terms of a geometric series. Find the 1st term and the constant difference of the arithmetic series ?

2. ### Math

An arithmetic series has first term -50 and common difference 4. How many terms are in the series so that the sum of the series first exceeds 100?

3. ### Math question

In an arithmetic series, the 15th term is 43 and the sum of the first 15 terms is 120. Add the first 20 terms of the series.

4. ### Calculus

By recognizing each series below as a Taylor series evaluated at a particular value of x, find the sum of each convergent series. A) 1+5 + (5^2)/(2!)+(5^3)/(3!)+(5^4)/(4!)+...+ (5^k)/(k!)+...= B)

1. ### math

in a geometric series t1=23,t3=92 and the sum of all of the terms of the series is 62 813. how many terms are in the series?

2. ### Calculus

Which one of the following statements is true about the series the series from n equals 1 to infinity of the quotient of negative 1 raised to the nth power and n ? (4 points) Is this A or B? I am a little confused. A) It is

3. ### calculus

Use multiplication of power series to find the first three non-zero terms of the Maclaurin series of e^x ln(1 − x).

4. ### calculus

Find the Maclaurin series of xcos(x^2).