series expansion for sin(e^x)

Use the series expansions for and sin(x) to find a series

expansion forsin( )up to terms in x .

Use your series to approximate http://quicklatex.com/cache3/ql_4184...e145775_l3.png

stating the error in your approximation.

ok i have the series expansion for

http://quicklatex.com/cache3/ql_f9cd...e50535a_l3.png

http://quicklatex.com/cache3/ql_19db...5567526_l3.png

how do i combine the two series to get sin(e^x).

also what does it mean by stating the error of approximation.

Re: series expansion for sin(e^x)

Substitute one into the other

or

Then use .

Although probably not the way I would do it.

Re: series expansion for sin(e^x)

Substitute therefore the serie expansion of . Replacing by gives:

EDIT:

Sorry Danny, I didn't see your post.

Re: series expansion for sin(e^x)

Re: series expansion for sin(e^x)

following sirons method I have done the following ( is it correct?)

http://quicklatex.com/cache3/ql_5434...308d737_l3.png

Re: series expansion for sin(e^x)

Re: series expansion for sin(e^x)

The OP required 'a series expansion for up to terms in ' and, in my opinion, the simplest way to achieve that result is to use the standard approach described in my previous post, that uses the derivatives of in to obtain...

(1)

Proceeding we have...

(2)

(3)

(4)

(5)

... so that the (1) becomes...

(6)

The 'exact values' of [in black...] and its values computed with (6) [in red...] for are represented in the figure...

http://digilander.libero.it/luposabatini/MHF128.bmp

The approximation is 'good' in the range but outside the error rapidly increases... in particular for the 'exact value' is and the value computed with (6) ...

Kind regards