Question

(a) Use the power series expansions for ex, sin x, cos x, and geometric series to find the first three nonzero terms in the power series expansion of the given function.

(b) Based on the information given in the section on algebraic properties of power series, for which values of x can you guarantee that the new series converges.
(If you have a CAS, you can easily find several more nonzero terms in the power series expansions of the functions.)
(e^x)/(cos(x))

Answer 1

a)
`\mathbf{4 + (x)/(1!)- (2x^2)/(2!) ...}`

b) See Below for proper explanation

Explanation:

a) The objective here is to Use the power series expansions for ex, sin x, cos x, and geometric series to find the first three nonzero terms in the power series expansion of the given function.

The function is
`e^x + 3 \ cos \ x`

The expansion is of
`e^x` is
`e^x = 1 + (x)/(1!)+ (x^2)/(2!)+ (x^3)/(3!) + ...`

The expansion of cos x is
`cos \ x = 1 - (x^2)/(2!)+ (x^4)/(4!)- (x^6)/(6!)+ ...`

Therefore;
`e^x + 3 \ cos \ x = 1 + (x)/(1!)+ (x^2)/(2!)+ (x^3)/(3!) + ... 3[1 - (x^2)/(2!)+ (x^4)/(4!)- (x^6)/(6!)+ ...]`

`e^x + 3 \ cos \ x = 4 + (x)/(1!)- (2x^2)/(2!) + (x^3)/(3!)+ ...`

Thus, the first three terms of the above series are:

`\mathbf{4 + (x)/(1!)- (2x^2)/(2!) ...}`

b)

The series for
`e^x + 3 \ cos \ x` is
`\sum \limits^(\infty)_(x=0) (x^x)/(n!) + 3 \sum \limits^(\infty)_(x=0) ( -1 )^x (x^(2x))/((2n)!)`

let consider the series;
`\sum \limits^(\infty)_(x=0) (x^x)/(n!)`

`|(a_x+1)/(a_x)| = | (x^(n+1))/((n+1)!) * (n!)/(x^x)| = |(x)/((n+1))| \to 0 \ as \ n \to \infty`

Thus it converges for all value of x

Let also consider the series
`\sum \limits^(\infty)_(x=0)(-1)^x(x^(2n))/((2n)!)`

It also converges for all values of x