There are standard canned Maclaurin Series (sin(x) , Cos(x), arctan(x) ,e to the x) that can be used when the a term in the taylor equation the derivative value times(x-a) to n over n factorial = 0 (turns taylor into def of maclaurin)The trouble is when they do not = 0 at least in the case of the trig functions another non zero term is generated.
So at pi/4 or it's equivalent at each quadrant the maximum value of the extra term is generated. A zero extra term is only generated
on the coordinate axis . This says to me these canned functions can only be used on the axis.
This also means in the case of sin and cos the alternating in alternating terms alternates every other term .
This all comes back to alternating series and taylor error/Remainder error.
And the definition of an alternating series,. A series that alternates every other term or multiple terms is it still alternating?
If that is the case does only the Taylor rule apply to it.
Also it looks like the canned Maclaurin functions cannot be used at all for if a in x-a is non zero.
So I am looking for verification that
1) Alternating series error can only be used if the series is maclaurin and at the axis where one or the other is zero.(at least in the case of sin and cos)
2)Official definition of alternation . (must be every other term or every n terms?)
3)When you can and when you can't use each approach for error.
4 ) These questions got started from the question use a Taylor Series to estimate
the cos of 69 degrees to 6 terms 23pi/60 ( 9 degrees is 3pi/20 plus 60 is 20pi/60 )
So at pi/4 or it's equivalent at each quadrant the maximum value of the extra term is generated. A zero extra term is only generated
on the coordinate axis . This says to me these canned functions can only be used on the axis.
This also means in the case of sin and cos the alternating in alternating terms alternates every other term .
This all comes back to alternating series and taylor error/Remainder error.
And the definition of an alternating series,. A series that alternates every other term or multiple terms is it still alternating?
If that is the case does only the Taylor rule apply to it.
Also it looks like the canned Maclaurin functions cannot be used at all for if a in x-a is non zero.
So I am looking for verification that
1) Alternating series error can only be used if the series is maclaurin and at the axis where one or the other is zero.(at least in the case of sin and cos)
2)Official definition of alternation . (must be every other term or every n terms?)
3)When you can and when you can't use each approach for error.
4 ) These questions got started from the question use a Taylor Series to estimate
the cos of 69 degrees to 6 terms 23pi/60 ( 9 degrees is 3pi/20 plus 60 is 20pi/60 )