Finding the Maclaurin Series for 1/cos x up to the Term in x4

When we need to find the Maclaurin series of a function, such as ( frac{1}{cos x} ), up to and including the term in ( x^4 ), we can follow a systematic approach. The Maclaurin series is a special case of the Taylor series, expanded around the point ( x 0 ).

Step-by-Step Calculation

The function ( frac{1}{cos x} ) is an even function, meaning its Maclaurin series will only contain terms with even powers of ( x ). The general form of the Maclaurin series for such a function is:

1 ax^2 bx^4 ...

Geometric Series Interpretation

To find the series, we start by expressing the given function in a form that allows us to use a known series expansion. Consider the geometric series:

(1 / (1 - s) 1 s s^2 s^3 ... where ( s 1 - cos x )

We know that:

( 1 - cos x x^2 / 2 - x^4 / 24 ... )

Therefore:

( s x^2 / 2 - x^4 / 24 ... )

Substituting ( s ) into the geometric series, we get:

( frac{1}{cos x} frac{1}{1 - (x^2 / 2 - x^4 / 24 ...)} )

Expanding this, we can write:

( frac{1}{cos x} left( frac{1}{1 - (x^2 / 2 - x^4 / 24)} right) left( 1 - (x^2 / 2 - x^4 / 24) right)^{-1} )

Using the series expansion for ( (1 - A)^{-1} approx 1 A A^2 A^3 ... ), we get:

( frac{1}{cos x} left( 1 (x^2 / 2 - x^4 / 24) (x^2 / 2 - x^4 / 24)^2 ... right) left( 1 - (x^2 / 2 - x^4 / 24) right) )

Matching the Series Terms

By expanding and simplifying, we can equate the coefficients of the corresponding powers of ( x )

The series can be written as:

( frac{1}{cos x} 1 frac{x^2}{2} frac{5x^4}{24} ... )

Computing Coefficients

To confirm our expansion, consider the following calculation:

( 1 left( 1 ax^2 bx^4 ... right) left( 1 - frac{x^2}{2} - frac{x^4}{24} ... right) )

In this expansion, we discard terms of ( x^6 ) or higher power, so we get:

( 1 1 - frac{x^2}{2} - frac{x^4}{24} (a - frac{1}{2})x^2 (b - frac{a}{2} - frac{1}{24})x^4 ... )

Equating the coefficients of ( x^2 ) and ( x^4 ) on both sides, we find:

( a - frac{1}{2} 0 ) implies ( a frac{1}{2} )

( b - frac{a}{2} - frac{1}{24} 0 ) implies ( b - frac{1}{4} - frac{1}{24} 0 ) which simplifies to ( b frac{5}{24} )

Conclusion

The Maclaurin series for ( frac{1}{cos x} ) up to and including the term in ( x^4 ) is:

( frac{1}{cos x} 1 frac{x^2}{2} frac{5x^4}{24} )

This method can be applied to various even functions to find their Maclaurin series expansions up to a specific term.