S2 := x - x ^ 2  2
S3 := S2 + x ^ 3  3
f := Log@x + 1D
Plot@8S2, f, S3<, 8x, -2, 2<D
2
-2
-1
1
-2
-4
-6
Needs@"PlotLegends`"D
2
2
MTH2301Lab4.nb
Plot@8S2, f, S3<, 8x, 0, 1<, PlotLegend ® 8"S2", "f", "S3"<D
0.8
0.6
0.4
S2
0.2
f
0.2
0.4
0.6
0.8
1.0
S3
S4 := S3 - x ^ 4  4
S5 := S4 + x ^ 5  5
Plot@8S4, f, S5<, 8x, 0, 1<, PlotLegend ® 8"S4", "f", "S5"<D
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
Series@f, 8x, 0, 10<D
x2
x-
x3
+
2
x4
-
3
x5
+
4
x6
-
5
x7
+
6
7
ff := 1  H1 - xL
SS1 = Series@ff, 8x, 0, 1<D
1 + x + O@xD2
PP1 = Normal@%D
1+x
x8
-
x9
+
8
x10
-
9
10
+ O@xD11
1.0
MTH2301Lab4.nb
Plot@8ff, PP1<, 8x, -1, 1<, PlotLegend ® 8"ff", "PP1"<D
3.5
3.0
2.5
2.0
1.5
1.0
ff
-1.0
0.5
-0.5
0.5
1.0
PP1
PP2 = Normal@Series@ff, 8x, 0, 2<DD
1 + x + x2
Plot@8ff, PP1, PP2<, 8x, -1, 1<, PlotLegend ® 8"ff", "PP1", "PP2"<D
3.5
3.0
2.5
2.0
1.5
ff
1.0
0.5
PP1
-1.0
-0.5
PP2
Pn10 = Normal@Series@ff, 8x, 0, 10<DD
1 + x + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10
0.5
1.0
3
4
MTH2301Lab4.nb
Plot@8ff, PP1, PP2, Pn10<, 8x, -1, 1<, PlotLegend ® 8"ff", "PP1", "PP2", "PP10"<D
4
3
2
ff
1
PP1
PP2
-1.0
-0.5
0.5
PP10
Series@Log@3 + xD, 8x, 0, 6<D
x2
x
Log@3D +
3
x3
+
18
x4
-
81
x5
+
324
x6
-
1215
+ O@xD7
4374
PPP1 = Normal@Series@Log@3 + xD, 8x, 0, 2<DD
x2
x
3
+ Log@3D
18
MTH2301Lab4.nb
Plot@8Log@3 + xD, PPP1<, 8x, -3, 10<, PlotLegend ® 8"Log", "PPP1"<D
2
1
-2
2
4
6
8
10
-1
Log
-2
PPP1
PPP2 = Normal@Series@Log@3 + xD, 8x, 0, 4<DD
x2
x
3
x3
+
18
x4
-
81
+ Log@3D
324
Plot@8Log@3 + xD, PPP1, PPP2<, 8x, -3, 10<, PlotLegend ® 8"Log", "PPP1", "PPP2"<D
2
1
-2
2
4
6
8
-1
Log
-2
PPP1
-3
PPP2
PPP8 = Normal@Series@Log@3 + xD, 8x, 0, 8<DD
x2
x
3
x3
+
18
x4
-
81
x5
+
324
x6
-
1215
x7
+
4374
x8
-
15 309
+ Log@3D
52 488
10
5
6
MTH2301Lab4.nb
Plot@8Log@3 + xD, PPP1, PPP2, PPP8<, 8x, -3, 10<, PlotLegend ® 8"Log", "PPP1", "PPP2", "PPP8"<D
2
-2
2
4
6
-2
Log
-4
PPP1
PPP2
-6
PPP8
PPP20 = Normal@Series@Log@3 + xD, 8x, 0, 20<DD
x2
x
x3
3
+
18
x4
-
81
x5
324
x11
x7
1215
x12
4374
x17
2 195 382 771
15 309
+
52 488
20 726 199
x18
6 973 568 802
177 147
+
66 961 566
x19
+
x10
x16
-
215 233 605
+
688 747 536
x20
-
22 082 967 873
+
590 490
x15
-
-
x9
x14
+
6 377 292
x8
+
x13
1 948 617
x6
+
+ Log@3D
69 735 688 020
8
MTH2301Lab4.nb
PPP40 = Normal@Series@Log@3 + xD, 8x, 0, 40<DD
x2
x
3
x3
x4
+
18
x5
81
x6
+
324
x13
4374
x14
15 309
x15
52 488
x24
x25
2 195 382 771
x23
690 383 311 398
x27
+
66 088 511 536 554
205 891 132 094 649
x30
x31
1 990 280 943 581 607
x32
+
6 176 733 962 839 470
x33
19 147 875 284 802 357
x34
+
59 296 646 043 258 912
2 165 293 113 021
x26
x29
640 550 188 738 908
+
6 973 568 802
+
-
+
183 448 998 696 332 259
x35
+
567 024 177 788 663 346
x36
-
1 751 104 078 464 989 745
x37
+
5 403 406 870 691 968 356
x38
16 660 504 517 966 902 431
x39
+
51 332 365 271 573 699 382
+
6 377 292
x18
x22
219 667 417 263
21 182 215 236 075
1 948 617
-
-
+
x28
590 490
-
x17
x21
69 735 688 020
x12
+
+
+
6 778 308 875 544
177 147
688 747 536
x20
x11
-
x16
215 233 605
-
x10
+
-
66 961 566
x19
x9
-
+
22 082 967 873
x8
+
1215
20 726 199
x7
-
x40
-
158 049 650 967 740 074 413
+ Log@3D
486 306 618 362 277 152 040
Plot@8Log@3 + xD, PPP1, PPP2, PPP8, PPP20, PPP40<, 8x, -2, 2<,
PlotLegend ® 8"Log", "PPP1", "PPP2", "PPP8", "PPP20", "PPP40"<D
1.5
1.0
Log
0.5
PPP1
PPP2
PPP8
PPP20-2
PPP40
-1
1
7