W9.2 Taylor Series
The Taylor series (sometimes known as the Maclaurin series) expresses a function f(x) as an ‘expansion,’ in powers of x, about x=0.
It is probably the most frequently-used mathematical tool in the physicist’s toolkit! This exercise is therefore particularly important .
- Review the online introduction to Taylor Series (embedded in S6.2)
- Establish the first two non-zero terms in the expansions
of the following functions:
- ex
- ln(1-y)
- tanθ
Solution
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Solution
Outline Answers
Let f(x)=ex
Then
Thus
ex=1+x+O(x2)If you haven’t met it before the notation
O(xm)is used to signify that the terms neglected in an approximation are of (at least) the m-th power in the variable x (which is presumed small).Let f(y)=ln(1-y)
Then
Thus
Let f(θ)=tanθ
Then
Thus
tanθ=θ+O(θ2)Let
Then
Thus
Let
Then
To pick up the second non-zero term we must therefore consider the second derivative:
ThusYou can establish this result a little more easily by showing first that
and then setting y≡x2