Colin Series...going off on a tangent!

hi were doing macluarin series and taylor expansions, only really maclaurin ones as were looking at derivatives at 0.

Anyhow, we did sinx and cosx but not tanx. And when i asked my teacher, he said its too hard and you wont need to know it.

So could anyone explain how to find the maclaurin series for tanx such that it is in closed form.

Or atleast a good method for working out each term rather than repeated differentiation which i have found to be very tiresome already after the 5th derivative. Is there a recurrence relation between the derivatives?

I understand the closed form involves Bernoulli's function so I assume I would not be able to find the general term with just FP2 knowledge?

Thanks

Reply 1

davros

Original post by abra-cad-abra

There are numerous approaches.

If you google "tangent maclaurin series" or "maclaurin series for tan x" you'll find numerous links to different ways :smile:

Reply 2

abra-cad-abra

Original post by davros

There are numerous approaches.

If you google "tangent maclaurin series" or "maclaurin series for tan x" you'll find numerous links to different ways :smile:

what way do you think could be approached with my knowledge. Something that a step question could be based off perhaps?

Reply 3

abra-cad-abra

Original post by davros

There are numerous approaches.

If you google "tangent maclaurin series" or "maclaurin series for tan x" you'll find numerous links to different ways :smile:

also if i differentiate a series. is it necessarily the differential of the sigma form of the sum

so if i differentiaed sinx. when looking at it in its maclaurin expansion i can differentiate the general term to get the general term of cosx expansion

is this true for any series? so d/dx SIGMA f(r) (r=0 to r=inf or n) is equivalent to SIGMA f'(r) (r=0 to r=inf or n) ???

Reply 4

Mr M

Study Forum Helper

Original post by abra-cad-abra

what way do you think could be approached with my knowledge. Something that a step question could be based off perhaps?

http://www.thestudentroom.co.uk/showthread.php?t=1019661

Reply 5

davros

Original post by abra-cad-abra

(Sorry, I'm not normally so lazy with suggestions but I've had a long day and turning in soon, so Google seemed the best answer for suggestions to be honest :smile:

)

Personally I would just use the repeated differentiation method to get the tangent series but there are other methods such as tan x = sin x / cos x so sin x = (cos x)(tan x) then you can multiply series and compare coefficients (like you do with polynomials!).

You can differentiate a series within its "radius of convergence" so a series like sin x that converges for all values of x can be differentiated to give you the cos x series etc.

You can also go the other way via integration, but obviously that introduces an arbitrary constant so you need a bit more information like the value of the function at 0 to pin down that constant :smile:

Reply 6

Blazy

Original post by abra-cad-abra

Or atleast a good method for working out each term rather than repeated differentiation which i have found to be very tiresome already after the 5th derivative.