Derivatives of any size (the hard case)

Now, let's look back at  .  We would like to approximate this function using a Taylor series over a particular interval.

We also need to go back to our complicated function to find the maximum size of the derivative.

Again, let's use the errorBoundFunction, etc. defined above.

Let's take a look at the error as a function of n again (this graph takes a long time to generate on my computer, so be warned unless you have a pretty fast computer):

(We had to use ListPlot instead of Plot because of our complicated maxNthDerivValue function.  It also caused all those error message, which are mostly harmless.)

Notice that the error bound gets extremely large before it starts to decline again for this function.  Also, all the error messages are indications where the FindMinimum function is having trouble.

In order to get our error bound within , it is going to take a lot of terms.  I don't think there is a reasonable way to find this automatically, so we will just make a table of error bounds starting with the degree and going up, finding the answer by inspection:

n	Maximum error bound
180	0.00007170903547160493`
181	0.00006841835913534768`
182	0.00003442152219036721`
183	3.291090338181753`*^-6
184	0.000018373002532366496`
185	0.00001574369459218812`
186	5.903851388224089`*^-6
187	5.4138505863447204`*^-6
188	6.526735322089658`*^-6
189	1.0315713798956777`*^-6
190	7.137703906566557`*^-7
191	8.745469520687992`*^-7
192	5.651662749310031`*^-7
193	2.184475744801437`*^-7
194	3.075868735955284`*^-8
195	2.1854762605026947`*^-9
196	5.976660464647574`*^-8
197	1.6206096630114732`*^-8
198	1.6403054849188125`*^-8
199	2.1931580417080432`*^-8
200	9.610386677871184`*^-9
201	1.2174663499238547`*^-9
202	3.489024193661968`*^-9
203	4.860702433839462`*^-9
204	2.2988018859513834`*^-9
205	1.3330197777394226`*^-9
206	1.5179259645505997`*^-9
207	5.450556775102266`*^-10
208	2.917247733079802`*^-10
209	6.188062008301917`*^-10
210	2.576359640187811`*^-11
211
212	2.0791723388638945`*^-11
213	1.994938127004691`*^-11
214	1.653600007562458`*^-11
215	1.8001833119583544`*^-11
216
217	1.2372317539621394`*^-11
218	4.672624529043278`*^-12
219	1.9284572100266505`*^-12
220	8.804557310078477`*^-13
221	3.3409374400649055`*^-13
222	3.452193029003685`*^-13
223	3.5066765647303556`*^-13
224
225	8.113377620687274`*^-14
226	3.9069780791491264`*^-14
227	5.843961329310069`*^-14
228	2.8839167203939646`*^-14
229	3.185406626599049`*^-15
230	1.2264722333865014`*^-14

So, it looks like it will take a degree Taylor series to achieve this degree of accuracy throughout the domain.  Let's double-check the derivative:

Did it work?

Something odd is definitely going on here.  What do you think the problem is and how would you suggest we attempt to fix it?