Sample expansions of 1/(1}x)            To the theory     To numerical studies
                          To the calculation form for 1/(1-x)  To the calculation form for 1/(1+x)

For example, as p=2 and q=5, we show expressions (1) and (16) in detail.iValue of ri varies by
expressions.j

First degree
1/(1-x)=-0.96+0.16x+r1@@@Valid in 2ƒxƒ5   (1)
1/(1+x)=-0.96-0.16x+r1      Valid in -5ƒxƒ-2@ (16)
1/(1-x)=0.39506+0.04938x+r1@@@Valid in -5ƒxƒ-2     (1)
1/(1+x)=0.39506-0.04938x+r1      Valid in  2ƒxƒ5@@ (16)

Second degree
1/(1-x)=-1.744+0.608x-0.064x2+r2@@@Valid in 2ƒxƒ5@@(1)
1/(1+x)=-1.744-0.608x-O.064x2+r2      Valid in -5ƒxƒ-2@@(16)
1/(1-x)=0.52949+0.12620x+0.01097x2+r2@@@Valid in -5ƒxƒ-2      (1)
1/(1+x)=0.52949-0.12620x+0.01097x2+r2      Valid in 2ƒxƒ5      (16)

Third degree
1/(1-x)=-2.8416+1.5488x-0.3328x2+0.0256x3+r3@@@Valid in 2ƒxƒ5     (1)
1/(1+x)=-2.8416-1.5488x-O.3328x2-0.0256x3+r3      Valid in -5ƒxƒ-2     (16)
1/(1-x)=0.63405+0.21582x+0.03658x2+O.00244x3+r3@@@ Valid in -5ƒxƒ-2      (1)
1/(1+x)=0.63405-0.21582x+0.03658x2-0.00244x3+r3       Valid in 2ƒxƒ5       (16)

Fourth degree
1/(1-x)=-4.37824+3.30496x-1.08544x2+0.16896x3-0.01024x4+r4@@ Valid in 2ƒxƒ5   (1)
1/(1+x)=-4.37824-3.30496x-1.08544x2-0.16896x3-0.01024x4+r4     Valid in -5ƒxƒ-2    (16)
1/(1-x)=0.71537+0.30876x+0.07641x2+O.01003x3+0.00054x4+r4@@@Valid in -5ƒxƒ-2     (1)
1/(1+x)=0.71537-0.30876x+0.07641x2-0.01003x3+0.00054x4+r4      Valid in  2ƒxƒ5     (16)

Effect of p and q
 When we fix n and (q+p)/2, the expressions do not change for the change of p and q. (q+p)/2
locates the center of the interval (p,q). If (q+p)/2, the center of the interval changes,
the expression changes. In case of 1/(1-x), when p(or q)is near to 1 and the width of the
interval is small, the coefficients have a tendency to be large. If we replace x to -x in
the expressions below, we get expressions for 1/(1+x), then the valid interval changes from
(p,q) to (-q,-p). Sample expressions of fourth degree are follows.iCoefficients in the
expressions of third degree or under will be different.)

Cases where p is far from 1.
1/(1-x)=-2.51336+1.39037x-0.34462x2+0.040935x3-0.0019040x4+r4     Valid in 3ƒxƒ6
1/(1-x)=0.63335+0.22597x+0.044905x2+0.0046693x3+0.00019869x4+r4     Vlid in 6ƒxƒ-3
1/(1-x)=-0.41194+0.058707x-0.0040459x2+0.00013759x3-0.0000018593x4+r4     Valid in 10ƒxƒ20
1/(1-x)=0.27580+0.034405x+0.0022182x2+0.000072479x3+0.00000095367x4+r4@@ Valid in -20ƒxƒ-10@

Cases where p is near to 1.
1/(1-x)=59050-269000x+460000x2-350000x3+100000x4+r4      Valid in 0.8ƒxƒ1
1/(1-x)=-161050+571000x-760000x2+450000x3-100000x4+r4@@  Valid in 1ƒxƒ1.2
1/(1-x)=2-8x+32x2-48x3+32x4+r4      Valid in 0ƒxƒ1
1/(1-x)=-242+568x-512x2+208x3-32x4+r4      Valid in 1ƒxƒ2
1/(1-x)=1+x+x2+x3+x4+r4       Valid in -1ƒxƒ1
1/(1-x)=0.99588+0.95473x+0.79012x2+0.46091x3+0.13169x4+r4      Vakid in -2ƒxƒ1
1/(1-x)=-6.59375+6.06250x-2.37500x2+0.43750x3-0.031250x4+r4     Vakid in 1ƒxƒ5
1/(1-x)=0.86831+0.53909x+0.20988x2+0.045267x3+0.0041152x4+r4      Valid in -5ƒxƒ1
1/(1-x)=-0.61051+0.12154x-0.011560x2+0.000540x3-0.000010x4+r4      Valid in 1ƒxƒ21
1/(1-x)=0.37908+0.068618x+0.0065259x2+0.00031667x3+0.0000062092x4+r4      Valid in -21ƒxƒ1