RevLimitSumSer.nb
H* evaluate limits, list terms of a sequence,
and plot terms of a sequence *LLimit@Sin@xD  x, x ® 0D
1
Limit@H1 + 1  nL ^ n, n ® InfinityD
ã
Limit@n !  5 ^ n, n ® ¥D
¥
00
1
encountered. ‡
Power::infy : Infinite expression
0
¥::indet : Indeterminate expression 0 ComplexInfinity encountered. ‡
Indeterminate
0 ^ Infinity
0
Table@1  n ^ 2, 8n, 1, 5<D
:1,
1
1
,
4
1
,
9
1
,
16
25
>
ListPlot@Table@1  n ^ 2, 8n, 1, 5<DD
1.0
0.8
0.6
0.4
1
2
3
4
5
H* Take derivatives of all orders. Find indefinite integrals evaluate definite integrals
exactly and approximately to any specified number of decimal places. *LD@1  x, xD
1
x2
1
DB , 8x, 2<F
x
2
x3
1
RevLimitSumSer.nb
1
DB , 8x, 3<F
x
6
x4
1
TableBDB , 8x, n<F, 8n, 0, 3<F
x
1
1
2
6
: ,,
,>
2
3
x
x
x
x4
Integrate@1  x ^ 2, xD
1
x
Integrate@1  x ^ 2, 8x, 1, 10<D
9
10
Integrate@1  H1 + x ^ 2L, xD
ArcTan@xD
Integrate@1  H1 + x ^ 2L, 8x, 1, 5<D
Π
-
+ ArcTan@5D
4
NIntegrate@1  H1 + x ^ 2L, 8x, 1, 5<D
0.588003
Integrate@1  x ^ 2, 8x, 1, Infinity<D
1
Integrate@1  H1 + x ^ 2L, 8x, - Infinity, Infinity<D
Π
Integrate@1  x ^ 2, 8x, 0, Infinity<D
1
Integrate::idiv : Integral of
x2
à
¥
0
does not converge on 80, ¥<. ‡
1
âx
x2
H* Find sum of finite and infinite series,
exactly and decimal approximation *LSum@1  n ^ 2, 8n, 1, Infinity<D
Π2
6
NSum@1  n ^ 2, 8n, 1, Infinity<D
1.64493
2
RevLimitSumSer.nb
Sum@1  n !, 8n, 0, Infinity<D
ã
N@Sum@1  n !, 8n, 0, Infinity<D, 20D
2.7182818284590452354
Sum@1  n, 8n, 1, Infinity<D
Sum::div : Sum does not converge. ‡
â
¥
1
n=1
n
H* Find power series of fHxL about x equal a up to order n
HTaylor seriesL. Find coefficient in power series*LSeries@f@xD, 8x, 0, 5<D
f@0D + f¢ @0D x +
1
2
f¢¢ @0D x2 +
1
6
fH3L @0D x3 +
1
24
fH4L @0D x4 +
1
120
fH5L @0D x5 + O@xD6
Normal@Series@f@xD, 8x, 0, 5<DD
f@0D + x f¢ @0D +
1
2
x2 f¢¢ @0D +
1
6
x3 fH3L @0D +
1
24
x4 fH4L @0D +
1
120
x5 fH5L @0D
SeriesCoefficient @f@xD, 8x, 0, 3<D
1
6
fH3L @0D
Series@Log@1 + xD, 8x, 0, 10<D
x2
x3
x-
+
x4
-
2
x5
+
3
4
x6
-
5
x7
x8
+
6
7
x9
+
8
x10
-
9
+ O@xD11
10
Series@E ^ x, 8x, 0, 10<D
x2
1+x+
x3
+
2
x4
+
6
x5
+
24
x6
+
120
x7
+
720
x8
+
5040
x9
x10
+
40 320
+
362 880
+ O@xD11
3 628 800
Series@E ^ x, 8x, 1, 10<D
ã + ã Hx - 1L +
1
720
1
2
ã Hx - 1L2 +
ã Hx - 1L6 +
ã Hx - 1L7
5040
1
6
+
ã Hx - 1L3 +
ã Hx - 1L8
40 320
+
1
24
ã Hx - 1L4 +
ã Hx - 1L9
362 880
+
1
120
ã Hx - 1L5 +
ã Hx - 1L10
+ O@x - 1D11
3 628 800
H* the easiest way to find the Maclaurin series of arctan x is to integrate term by term
from 0 to x the Maclaurin series of 1H1+t^2 *L Normal@Series@ArcTan@xD, 8x, 0, 10<DD
x3
x-
x5
+
3
x7
-
5
x9
+
7
9
Integrate@Normal@Series@1  H1 + x ^ 2L, 8x, 0, 9<DD, xD
x3
x-
x5
+
3
x7
-
5
x9
+
7
9
3
RevLimitSumSer.nb
Normal@Series@E ^ x, 8x, 0, 10<DD
x2
1+x+
x3
+
2
x4
+
6
x5
+
24
x6
+
120
x7
+
720
x8
+
5040
x9
+
40 320
x10
+
362 880
3 628 800
Normal@Series@E ^ x, 8x, 0, 10<DD
x2
1+x+
x3
+
2
x4
+
6
x5
+
24
x6
+
120
x7
+
720
x8
+
5040
x9
+
40 320
x10
+
362 880
3 628 800
H* The Maclaurin series of e^x is an alternating series when x equals
minus 1. The series converges to 1e. The magnitude of the Hn+1Lst term
is 1n! For three decimal place accuracy of the estimation of 1e require
H1n!<.0005L and truncate the series at the nth term. Make a table. *L
N@Table@1  n !, 8n, 1, 10<DD
91., 0.5, 0.166667, 0.0416667, 0.00833333, 0.00138889,
0.000198413, 0.0000248016 , 2.75573 ´ 10-6 , 2.75573 ´ 10-7 =
H* For 3 place accuracy need up to 7th term, 1720 *L
NSum@H- 1L ^ n  n !, 8n, 0, 6<D
0.368056
H* The mathematica value of 1e is given below. The truncated series
estimate is accurate to 3 decimal places but not 4 decimal places. If fewer
than 7 terms are taken the estimate is not accurate to 3 decimal places *L
N@
1  ED
0.367879
NSum@H- 1L ^ n  n !, 8n, 0, 5<D
0.366667
H* Use the Floor function for signs alternating in groups. The floor of x
is the greatest integer less than or equal to x. Use H-1L^n for alternating
signs and H-1L^Floor@nkD for signs alternating in groups of k *LFloor@0D
0
Floor@1  2D
0
Floor@1D
1
Floor@3  2D
1
Table@Floor@n  2D, 8n, 0, 3<D
80, 0, 1, 1<
4
RevLimitSumSer.nb
Table@Floor@n  3D, 8n, 0, 8<D
80, 0, 0, 1, 1, 1, 2, 2, 2<
Table@Floor@n  4D, 8n, 0, 11<D
80, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2<
Table@H- 1L ^ HFloor@n  2DL, 8n, 0, 3<D
81, 1, - 1, - 1<
Table@H- 1L ^ Floor@n  3D, 8n, 0, 8<D
81, 1, 1, - 1, - 1, - 1, 1, 1, 1<
Table@H- 1L ^ Floor@n  4D, 8n, 0, 11<D
81, 1, 1, 1, - 1, - 1, - 1, - 1, 1, 1, 1, 1<
H* Find binomial series and binomial coefficients *LSeries@H1 + xL ^ H1  2L, 8x, 0, 10<D
x2
x
1+
2
x3
+
8
5 x4
-
16
7 x5
+
128
21 x6
-
256
33 x7
+
1024
429 x8
715 x9
2048
2431 x10
+
32 768
65 536
+ O@xD11
262 144
Binomial@1  2, 3D
1
16
H1  2L H- 1  2L H- 3  2L  3 !
1
16
Series@H1 + xL ^ H5  2L, 8x, 0, 10<D
15 x2
5x
1+
+
2
5 x3
+
8
5 x4
-
16
3 x5
+
128
5 x6
-
256
5 x7
+
1024
45 x8
-
2048
32 768
Binomial@5  2, 6D
5
1024
H5  2L H3  2L H1  2L H- 1  2L H- 3  2L H- 5  2L  6 !
5
1024
55 x9
+
143 x10
-
65 536
262 144
+ O@xD11
5