Substitute one into the other
or
Then use .
Although probably not the way I would do it.
Use the series expansions for and sin(x) to find a series
expansion forsin( )up to terms in x .
Use your series to approximate
stating the error in your approximation.
ok i have the series expansion for
how do i combine the two series to get sin(e^x).
also what does it mean by stating the error of approximation.
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...
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