Bounds on remainder term of power series of elementary functions This is mainly a question about the remainder term of power series for elementary functions.
I'm very interested in aspects of calculating or computing elementary operations and functions, by which I mean:


*

*trigonometric: $\sin$, $\cos$, $\tan$

*inverse trig.: $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$

*log and exponential: $\ln$, $\exp$

*hyperbolic: $\sinh$, $\cosh$, $\tanh$

*inverse hyp.: $\sinh^{-1}$, $\cosh^{-1}$, $\tanh^{-1}$

*powers, reciprocation, $\sqrt{\ \ \ }$


perhaps also:


*

*gamma function: $\Gamma$

*and a few other important functions


There are many contexts (of calculation). For example:


*

*real versus complex arguments

*known, fixed precision versus variable precision

*numerical versus symbolic


There are many approaches and techniques available too. For example:


*

*power series expansions and polynomial approximations

*use of relationships between the functions

*use of periodic or similar properties to shrink the domain

*lookup tables and interpolation

*CORDIC (used within some hand calculators I believe)

*exact methods

*interval or other error-tracking methods


Some good references to certain aspects include:


*

*Digital Library of Mathematical Functions: Elementary Functions

*Chee-Keng Yap, Fundamental problems of algorithmic algebra

*Behrooz Parhami, Computer Arithmetic: Algorithms and Hardware Designs
The main gap in my knowledge is in finding bounds for the error or remainder term in partial power series expansions of certain of the above functions. Some are fairly simple to determine, whilst others seem to be awkward.
Any pointers on this matter would be much appreciated.
Likewise for any further references on any other aspects of or techniques for calculating elementary functions.
 A: These bounds you are looking for can be obtained from majorant series.  What you seek is implemented in the Dynamic Dictionary of Mathematical Functions; Bruno Salvy gave a very nice talk on this topic at CICM 2010 in early July.
For the guaranteed numerics aspect, the expert is Marc Mezzarobba, a PhD student of Bruno's.  On that page, see the links to NumGfun (software, presentation and paper) for all the details you would ever want on the topic.
A: 
finding bounds for the error or remainder term in partial power series expansions

I think you want the Euler-Maclaurin Summation formula.  That bounds the remainder terms, although it would require knowing the closed form of the integral representation of the function you are calculating.  
$ \sum_{n=a}^b f(n) \sim \int_a^b f(x)\,dx + \frac{f(a)+f(b)}{2} + \sum_{k=1}^\infty \,\frac{B_{2k}}{(2k)!}\left(f^{(2k-1)}(b)-f^{(2k-1)}(a)\right) $
The paper by Apostol "Elementary view of Euler-Maclaurin" AMM vol 106 (1999) pp. 409-418  is very accessible.  
The following papers/books may also be helpful 


*

*R.P. Boas "Estimating
Remainders." Math. Mag. 51,
pp 83-89, (1978)

*http://www.tricki.org/article/Estimating_sums

*Bridger and Frampton Bounding Power
Series Remainders Math. Mag. 71 (1998), pp. 204-207

*Sofo. Computational Techniques for
the Summation of Series

*Ross.  Methods of Summation

*Davis.  Summation of Series.
