Download Math 113 Rate of Convergence or Divergence for Infinite Series

Math 113
Rate of Convergence or Divergence for Infinite Series
Chapter 11
The topic of Infinite Series is an important topic for a standard second semester Calculus course. Convergence and
divergence of series are usually discussed, along with power series, Taylor series, Maclaurin series, and perhaps other
types of series. One topic that is rarely discussed is the rate of convergence or divergence. This handout gives tables
describing the rate of convergence or divergence for some common series. For the convergence rate, the table is set out
as describing the actual sum, along with how many terms are required for the error to be less than a specified amount.
The tables given are derived from R. P. Boas, Jr. in his article “Partial Sums of Infinite Series, and How They Grow”
from American Mathematical Monthly, Volume 84, Issue 4, pp. 237-258, and also the article by R.P. Boas, Jr., and J.
W. Wrench, Jr., “Partial Sums of the Harmonic Series” from American Mathematical Monthly, Volume 78, Issue 8, pp.
864-870.
1.
2.
∞
X
1
n ln n(ln ln n)2
n=3
∞
X
1
n(ln
n)2
n=2
Series
1
2
4(s = 1.1)
4(s = 1.5)
3
4(s = 2)
5
4(s = 10)
4(s = 100)
6(r = .9)
6(r = .5)
6(r = .1)
7
8
9(x = .9)
9(x = .5)
9(x = .1)
10
3.
4.
Sum
38.406768
2.109743
10.584448
2.612375
0.937548
1.644934
0.605522
1.000995
1+8×10-31
10
2
10
9
1.718282
1.291286
2.230273
0.564468
0.001000
1.062500
∞
X
ln n
n2
n=2
∞
X
1
s
n
n=1
5.
6.
∞
X
1
2
n ln n
n=2
∞
X
rn
n=0
7.
8.
∞
X
1
n!
n=1
∞
X
1
n
n
n=1
9.
∞
X
2
xn
n=1
10.
∞
X
n=1
Number of terms required to calculate
the sum with error less than 1/2 times:
10−2
10−10
10−100
10−1000
T (3.14 × 1086 )
T4 (1)
T4 (100)
T4 (1000)
86
9
7.23 × 10
T (8.7 × 10 )
T2 (100)
T2 (1000)
T (33)
T (113)
T (1013)
T (10013)
160000
1.6 × 1021
1.6 × 10201 1.6 × 102001
1685
5.47 × 1011
4.7 × 10102 4.6 × 101003
10
200
2 × 10
2 × 10100
2 × 101000
42
T (9)
T (98)
T (997)
1
11
1.1 × 1011
1.1 × 10111
1
1
10
1.21 × 1010
73
247
2214
21883
9
36
335
3324
3
11
101
1001
5
13
70
450
3
10
57
386
7
15
46
147
2
5
18
57
1
3
10
31
2
2
3
34
Note: T (x) = T1 (x) = 10x. Tn (x) = T (Tn−1 (x)).
Table 1: Rate of Convergence
n
n−n
Rate of Convergence or Divergence for Infinite Series, page 2
For a divergent series, the number of terms before the partial sums pass a particular sum can be computed. In order to
easily compute the partial sums of a divergent series, the constant γ is needed, which is unique to each series. The
constant γ is defined as
( n
)
Z n
X
γ = lim
f (k) −
f (t)dt
n→∞
1
k=1
The partial sum, sn , of a series is approximately
Z
sn =
n
1
1
f (t)dt − f (n) + γ − Rn ,
2
for some remainder term Rn . Table 2 gives a small list of different divergent series and the number of terms required
to make the sum greater than a particular integer.
1.
2.
∞
X
1
ln(ln n)
n=3
∞
X
1
n=2
Series
Value of γ
Number of terms required to
make the sum greater than
3
ln n
3.
4.
∞
X
1
√
n
n=1
∞
X
ln n
n=2
n
∞
X
1
n
n=1
∞
X
1
6.
n
ln
n
n=1
5.
7.
∞
X
1
n ln n ln ln n
n=3
1
2
3
4
5
6
7
7.21848
0.80193
0.539645
-0.0728158
0.577216
0.428166
2.299927
1
3
5
12
11
8717
1
10
3
1.3 × 1029
56
4
1
5
7
17
31
5
1
7
10
24
83
6
1
9
14
33
227
7
1
12
18
43
616
10
1
20
33
89
12367
20
6
56
115
565
100
112
489
2574
1000
1812
7764
250731
1.39 × 106
2.72 × 108
1.53 × 1043
1000000
2.62 × 106
1.55 × 107
2.50 × 1011
1.53 × 10614
T (4.3 × 105 )
2.65 × 1019
1.1 × 10434
Note: T (x) = T1 (x) = 10x . Tn (x) = T (Tn−1 (x)).
Table 2: Rates of Divergence
5.1 × 10
1.4 × 1079
3.1 × 1019
1.4 × 10215
T (1.3 × 104 )
T (108)
T2 (2 × 106 )
1.6 × 104321
T (.7 × 1090
T (5 × 1042 )
T2 (1.1 × 1041 )
T2 (4.3 × 105 )
T3 (4.3 × 105 )
T (4 × 10433 )
T2 (8 × 10431 )
Rate of Convergence or Divergence for Infinite Series, page 3
The partial sums of the harmonic series warrant closer scrutiny since it is a central part of a calculus course. Table 3
gives the partial sums of the harmonic series before and after a particular sum is passed. It should be noted that since
the series diverges, this table could be extended indefinitely (There will always be a finite number of terms needed to
have the partial sums exceed a particular sum). Note that it takes over 6 trillion terms before the sum of 30 is reached!
2000000
However, it pales in comparison to series 7 in table 2. That series takes 1010
terms before it reaches a sum of 20!
That’s 1 followed by 10200000 zeros!
n
1
2
3
4
10
11
30
31
82
83
226
227
615
616
1673
1674
4549
4550
12366
12367
33616
33617
91379
91380
248396
248397
675213
675214
1835420
1835421
sn
1.0000000
1.5000000
1.8333333
2.0833333
2.9289682
3.0198773
3.9949871
4.0272451
4.9900200
5.0020682
5.9999614
6.0043667
6.9996507
7.0012740
7.9998882
8.0004855
8.9999882
9.0002080
9.9999621
10.0000430
10.9999879
11.0000177
11.9999921
12.0000030
12.9999972
13.0000012
13.9999998
14.0000013
14.9999998
15.0000003
0000000
0000000
3333333
3333333
5396825
4487734
3092039
9543652
7990908
7268016
2200195
0834556
2051078
9713416
0043066
7199578
8271136
6293114
4792164
0827580
6178215
0863642
0835199
5166563
0366726
2948078
8104146
6205340
3343363
7826776
n
4989190
4989191
13562026
13562027
36865411
36865412
100210580
100210581
272400599
272400600
740461600
740461601
2012783314
2012783315
5471312309
5471312310
14872568830
14872568831
40427833595
40427833596
109894245428
109894245429
298723530400
298723530401
812014744421
812014744422
2207284924202
2207284924203
6000022499692
6000022499693
sn
15.9999998
16.0000000
16.9999999
17.0000000
17.9999999
18.0000000
18.9999999
19.0000000
19.9999999
20.0000000
20.9999999
21.0000000
21.9999999
22.0000000
22.9999999
23.0000000
23.9999999
24.0000000
24.9999999
25.0000000
25.9999999
26.0000000
26.9999999
27.0000000
27.9999999
28.0000000
28.9999999
29.0000000
29.9999999
30.0000000
Table 3: Partial Sums of the Harmonic Series
9502053
9545382
4111463
1484992
7659423
0371993
9975431
0973330
9794638
0161744
9905082
0040133
9964162
0013844
9983702
0001979
9995548
0002272
9998134
0000608
9999850
0000760
9999847
0000182
9999932
0000055
9999964
0000010
9999985
0000002