Cyosis said:The power series of the exponential, [itex]e^x[/itex], is [itex]\sum_{n=0}^\infty x^n/n![/itex].
So if you have [itex]e^{-x}[/itex] you can compute the power series by substituting [itex]x \rightarrow -x[/itex] into the power series. Try it out.
I don't understand your second question. Are you asking for the power series of the sine and cosine? Or for the complex exponential representation?
Cyosis said:It's called the series expansion of the sine/cosine or the power series of the sine/cosine. I would suggest memorizing/deriving the series expansions for the more common functions.
Check this http://en.wikipedia.org/wiki/Taylor_seriesp out for a list of series expansions.
An exponential function is a mathematical function in which the variable appears in the exponent. It is written in the form y = a^x, where a is a constant and x is the variable.
The expansion of e^x is an infinite series that represents the value of the exponential function e^x. It is written as 1 + x + (x^2 / 2!) + (x^3 / 3!) + ... = ∑(x^n / n!).
To solve an exponential function, you can use logarithms or take the natural log (ln) of both sides. You can also use the expansion of e^x to find the value of the function at a specific x-value.
The expansion of sin x and cos x is a series of terms that represent the values of these trigonometric functions. It is written as sin x = x - (x^3 / 3!) + (x^5 / 5!) - (x^7 / 7!) + ... = ∑((-1)^n * (x^(2n+1) / (2n+1)!)) and cos x = 1 - (x^2 / 2!) + (x^4 / 4!) - (x^6 / 6!) + ... = ∑((-1)^n * (x^(2n) / (2n)!)).
The expansion of e^x and sin/cos x can be used to approximate values of these functions at specific x-values. It can also be used to solve differential equations and find the values of integrals involving these functions.