Why doesn’t the “Maclaurin/ Taylor series of tanx” have a pattern?
First of all, just to review the concepts of Maclaurin and Taylor series, I am giving the definitions below.
Maclaurin Series: If a function f can be differentiated n times, at x=0, then we define the nth Maclaurin polynomial for f to be:
Taylor Series: If a function f can be differentiated n times, at x=a, then we define the nth Taylor polynomial for f to be:
Basically, these two kinds of polynomials are used to replace trigonometric, exponential or any other geometrical functions because sometimes, it’s really hard to work with these functions in the computer.
If we see the Maclaurin/ Taylor series of sinx or cosx, we can easily identify a pattern from them for nth term. See the picture below:
Though, tanx equals to the ratio of sinx and cosx, it’s polynomial doesn’t have a pattern for the nth term. We cannot assume from this what will come after a certain number. If you want to know it, then you have to differentiate n times for nth terms like this:
So, we can see that each time, we are getting a different number after differentiating at 0. Finally, if you do it for 7th times, you get this equation.
We cannot identify a pattern from this. Now the question is why. From the unit circle approach,we can see that each time tanx is occurring at the odd multiples of 90 degree, it is becoming something like is “ 1/0” or “-1/0” which this undefined. So, tanx is discontinuous at the odd multiples of 90 degree. Therefore, some terms are missing and we cannot relate the previous terms with the next terms.
So, it’s better to avoid the Maclaurine or Taylor series of tanx because you have to differentiate it every time at a point if you want to know the next which is really time-consuming.
References:
1. Calculus: Early Transcendentals, Single Variable, 9th Edition
9780470182048
John Wiley & Sons (2009)
Anton, Bivens, Davis
2. Sullivan Precalculus, 9th Edition
9780321716835
Pearson Prentice Hall (2012)
Sullivan
