Summing an infinite series paradox

In an effort to get the "Easy Math Puzzles" thread back to truly easy puzzles, I'm making a separate thread for this.

Andy claims the following:

(1/1) - (1/2) + (1/3) - (1/4) + (1/5) - ... = ln(2).

He shows in a spreadsheet the sum of the series approaching ln(2) as the number of terms get larger. Here are just the first 200 terms.

index	Valule	Sum of series
1	1.000000	1.000000
2	-0.500000	0.500000
3	0.333333	0.833333
4	-0.250000	0.583333
5	0.200000	0.783333
6	-0.166667	0.616667
7	0.142857	0.759524
8	-0.125000	0.634524
9	0.111111	0.745635
10	-0.100000	0.645635
11	0.090909	0.736544
12	-0.083333	0.653211
13	0.076923	0.730134
14	-0.071429	0.658705
15	0.066667	0.725372
16	-0.062500	0.662872
17	0.058824	0.721695
18	-0.055556	0.666140
19	0.052632	0.718771
20	-0.050000	0.668771
21	0.047619	0.716390
22	-0.045455	0.670936
23	0.043478	0.714414
24	-0.041667	0.672747
25	0.040000	0.712747
26	-0.038462	0.674286
27	0.037037	0.711323
28	-0.035714	0.675609
29	0.034483	0.710091
30	-0.033333	0.676758
31	0.032258	0.709016
32	-0.031250	0.677766
33	0.030303	0.708069
34	-0.029412	0.678657
35	0.028571	0.707229
36	-0.027778	0.679451
37	0.027027	0.706478
38	-0.026316	0.680162
39	0.025641	0.705803
40	-0.025000	0.680803
41	0.024390	0.705194
42	-0.023810	0.681384
43	0.023256	0.704640
44	-0.022727	0.681913
45	0.022222	0.704135
46	-0.021739	0.682396
47	0.021277	0.703672
48	-0.020833	0.682839
49	0.020408	0.703247
50	-0.020000	0.683247
51	0.019608	0.702855
52	-0.019231	0.683624
53	0.018868	0.702492
54	-0.018519	0.683974
55	0.018182	0.702155
56	-0.017857	0.684298
57	0.017544	0.701842
58	-0.017241	0.684601
59	0.016949	0.701550
60	-0.016667	0.684883
61	0.016393	0.701277
62	-0.016129	0.685148
63	0.015873	0.701021
64	-0.015625	0.685396
65	0.015385	0.700780
66	-0.015152	0.685629
67	0.014925	0.700554
68	-0.014706	0.685848
69	0.014493	0.700341
70	-0.014286	0.686055
71	0.014085	0.700140
72	-0.013889	0.686251
73	0.013699	0.699950
74	-0.013514	0.686436
75	0.013333	0.699769
76	-0.013158	0.686612
77	0.012987	0.699599
78	-0.012821	0.686778
79	0.012658	0.699436
80	-0.012500	0.686936
81	0.012346	0.699282
82	-0.012195	0.687087
83	0.012048	0.699135
84	-0.011905	0.687230
85	0.011765	0.698995
86	-0.011628	0.687367
87	0.011494	0.698861
88	-0.011364	0.687498
89	0.011236	0.698734
90	-0.011111	0.687622
91	0.010989	0.698611
92	-0.010870	0.687742
93	0.010753	0.698495
94	-0.010638	0.687856
95	0.010526	0.698383
96	-0.010417	0.687966
97	0.010309	0.698275
98	-0.010204	0.688071
99	0.010101	0.698172
100	-0.010000	0.688172
101	0.009901	0.698073
102	-0.009804	0.688269
103	0.009709	0.697978
104	-0.009615	0.688363
105	0.009524	0.697886
106	-0.009434	0.688452
107	0.009346	0.697798
108	-0.009259	0.688539
109	0.009174	0.697713
110	-0.009091	0.688622
111	0.009009	0.697631
112	-0.008929	0.688703
113	0.008850	0.697552
114	-0.008772	0.688780
115	0.008696	0.697476
116	-0.008621	0.688855
117	0.008547	0.697402
118	-0.008475	0.688928
119	0.008403	0.697331
120	-0.008333	0.688998
121	0.008264	0.697262
122	-0.008197	0.689066
123	0.008130	0.697196
124	-0.008065	0.689131
125	0.008000	0.697131
126	-0.007937	0.689195
127	0.007874	0.697069
128	-0.007813	0.689256
129	0.007752	0.697008
130	-0.007692	0.689316
131	0.007634	0.696949
132	-0.007576	0.689374
133	0.007519	0.696892
134	-0.007463	0.689430
135	0.007407	0.696837
136	-0.007353	0.689484
137	0.007299	0.696783
138	-0.007246	0.689537
139	0.007194	0.696731
140	-0.007143	0.689589
141	0.007092	0.696681
142	-0.007042	0.689638
143	0.006993	0.696631
144	-0.006944	0.689687
145	0.006897	0.696584
146	-0.006849	0.689734
147	0.006803	0.696537
148	-0.006757	0.689780
149	0.006711	0.696492
150	-0.006667	0.689825
151	0.006623	0.696447
152	-0.006579	0.689869
153	0.006536	0.696404
154	-0.006494	0.689911
155	0.006452	0.696363
156	-0.006410	0.689952
157	0.006369	0.696322
158	-0.006329	0.689993
159	0.006289	0.696282
160	-0.006250	0.690032
161	0.006211	0.696243
162	-0.006173	0.690070
163	0.006135	0.696205
164	-0.006098	0.690108
165	0.006061	0.696168
166	-0.006024	0.690144
167	0.005988	0.696132
168	-0.005952	0.690180
169	0.005917	0.696097
170	-0.005882	0.690215
171	0.005848	0.696063
172	-0.005814	0.690249
173	0.005780	0.696029
174	-0.005747	0.690282
175	0.005714	0.695996
176	-0.005682	0.690314
177	0.005650	0.695964
178	-0.005618	0.690346
179	0.005587	0.695933
180	-0.005556	0.690377
181	0.005525	0.695902
182	-0.005495	0.690407
183	0.005464	0.695872
184	-0.005435	0.690437
185	0.005405	0.695843
186	-0.005376	0.690466
187	0.005348	0.695814
188	-0.005319	0.690495
189	0.005291	0.695786
190	-0.005263	0.690523
191	0.005236	0.695758
192	-0.005208	0.690550
193	0.005181	0.695731
194	-0.005155	0.690577
195	0.005128	0.695705
196	-0.005102	0.690603
197	0.005076	0.695679
198	-0.005051	0.690628
199	0.005025	0.695653
200	-0.005000	0.690653

By the way, ln(2) = 0.693147181. This series clearly is slow to converge.

Bob, however, argues the sum of the series is (3/2)*ln(2). His argument relies on reordering the terms. Alternating two positive terms from Andy's series and then one negative. So, his first 200 terms look like this:

index	Valule	Sum of series
1	1.000000	1.000000
2	0.333333	1.333333
3	-0.500000	0.833333
4	0.200000	1.033333
5	0.142857	1.176190
6	-0.250000	0.926190
7	0.111111	1.037302
8	0.090909	1.128211
9	-0.166667	0.961544
10	0.076923	1.038467
11	0.066667	1.105134
12	-0.125000	0.980134
13	0.058824	1.038957
14	0.052632	1.091589
15	-0.100000	0.991589
16	0.047619	1.039208
17	0.043478	1.082686
18	-0.083333	0.999353
19	0.040000	1.039353
20	0.037037	1.076390
21	-0.071429	1.004961
22	0.034483	1.039444
23	0.032258	1.071702
24	-0.062500	1.009202
25	0.030303	1.039505
26	0.028571	1.068077
27	-0.055556	1.012521
28	0.027027	1.039548
29	0.025641	1.065189
30	-0.050000	1.015189
31	0.024390	1.039579
32	0.023256	1.062835
33	-0.045455	1.017381
34	0.022222	1.039603
35	0.021277	1.060879
36	-0.041667	1.019213
37	0.020408	1.039621
38	0.019608	1.059229
39	-0.038462	1.020767
40	0.018868	1.039635
41	0.018182	1.057817
42	-0.035714	1.022103
43	0.017544	1.039647
44	0.016949	1.056596
45	-0.033333	1.023262
46	0.016393	1.039656
47	0.015873	1.055529
48	-0.031250	1.024279
49	0.015385	1.039663
50	0.014925	1.054589
51	-0.029412	1.025177
52	0.014493	1.039670
53	0.014085	1.053754
54	-0.027778	1.025977
55	0.013699	1.039675
56	0.013333	1.053008
57	-0.026316	1.026693
58	0.012987	1.039680
59	0.012658	1.052338
60	-0.025000	1.027338
61	0.012346	1.039684
62	0.012048	1.051732
63	-0.023810	1.027922
64	0.011765	1.039687
65	0.011494	1.051181
66	-0.022727	1.028454
67	0.011236	1.039690
68	0.010989	1.050679
69	-0.021739	1.028940
70	0.010753	1.039692
71	0.010526	1.050219
72	-0.020833	1.029385
73	0.010309	1.039695
74	0.010101	1.049796
75	-0.020000	1.029796
76	0.009901	1.039697
77	0.009709	1.049405
78	-0.019231	1.030175
79	0.009524	1.039699
80	0.009346	1.049044
81	-0.018519	1.030526
82	0.009174	1.039700
83	0.009009	1.048709
84	-0.017857	1.030852
85	0.008850	1.039702
86	0.008696	1.048397
87	-0.017241	1.031156
88	0.008547	1.039703
89	0.008403	1.048106
90	-0.016667	1.031440
91	0.008264	1.039704
92	0.008130	1.047834
93	-0.016129	1.031705
94	0.008000	1.039705
95	0.007874	1.047579
96	-0.015625	1.031954
97	0.007752	1.039706
98	0.007634	1.047340
99	-0.015152	1.032188
100	0.007519	1.039707
101	0.007407	1.047114
102	-0.014706	1.032408
103	0.007299	1.039708
104	0.007194	1.046902
105	-0.014286	1.032616
106	0.007092	1.039708
107	0.006993	1.046701
108	-0.013889	1.032812
109	0.006897	1.039709
110	0.006803	1.046512
111	-0.013514	1.032998
112	0.006711	1.039710
113	0.006623	1.046332
114	-0.013158	1.033174
115	0.006536	1.039710
116	0.006452	1.046162
117	-0.012821	1.033341
118	0.006369	1.039711
119	0.006289	1.046000
120	-0.012500	1.033500
121	0.006211	1.039711
122	0.006135	1.045846
123	-0.012195	1.033651
124	0.006061	1.039712
125	0.005988	1.045700
126	-0.011905	1.033795
127	0.005917	1.039712
128	0.005848	1.045560
129	-0.011628	1.033932
130	0.005780	1.039713
131	0.005714	1.045427
132	-0.011364	1.034063
133	0.005650	1.039713
134	0.005587	1.045299
135	-0.011111	1.034188
136	0.005525	1.039713
137	0.005464	1.045178
138	-0.010870	1.034308
139	0.005405	1.039714
140	0.005348	1.045061
141	-0.010638	1.034423
142	0.005291	1.039714
143	0.005236	1.044949
144	-0.010417	1.034533
145	0.005181	1.039714
146	0.005128	1.044842
147	-0.010204	1.034638
148	0.005076	1.039714
149	0.005025	1.044740
150	-0.010000	1.034740
151	0.004975	1.039715
152	0.004926	1.044641
153	-0.009804	1.034837
154	0.004878	1.039715
155	0.004831	1.044546
156	-0.009615	1.034930
157	0.004785	1.039715
158	0.004739	1.044454
159	-0.009434	1.035020
160	0.004695	1.039715
161	0.004651	1.044366
162	-0.009259	1.035107
163	0.004608	1.039716
164	0.004566	1.044282
165	-0.009091	1.035191
166	0.004525	1.039716
167	0.004484	1.044200
168	-0.008929	1.035271
169	0.004444	1.039716
170	0.004405	1.044121
171	-0.008772	1.035349
172	0.004367	1.039716
173	0.004329	1.044045
174	-0.008621	1.035424
175	0.004292	1.039716
176	0.004255	1.043972
177	-0.008475	1.035497
178	0.004219	1.039716
179	0.004184	1.043900
180	-0.008333	1.035567
181	0.004149	1.039717
182	0.004115	1.043832
183	-0.008197	1.035635
184	0.004082	1.039717
185	0.004049	1.043765
186	-0.008065	1.035701
187	0.004016	1.039717
188	0.003984	1.043701
189	-0.007937	1.035764
190	0.003953	1.039717
191	0.003922	1.043638
192	-0.007813	1.035826
193	0.003891	1.039717
194	0.003861	1.043578
195	-0.007692	1.035886
196	0.003831	1.039717
197	0.003802	1.043519
198	-0.007576	1.035944
199	0.003774	1.039717
200	0.003745	1.043463

By the way, (3/2)*ln(2) = 1.039720771. Again slow to converge.

Then, Charlie (not to be confused with our Charlie) comes along and says the sum of the series is infinity - infinity = can't be defined. His argument is to separate the positive and negative terms.

Positive terms: 1/1 + 1/3 + 1/5 + 1/7 + 1/9 + .... Nobody argues that this series converges to infinity.
Negative terms: -1/2 - 1/4 - 1/6 - 1/8 - 1/10 - .... Nobody argues that this series converges to negative infinity.

infinity - infinity = ?.

Who is right. If you say Adam, why are Bob and Charlie wrong?

The question for the poll is who do you think is right?

Please don't just throw out a link to a YouTube video. Rather, make your argument in YOUR words.

Good answer, Don, but what about Bob's argument?

Quote: Wizard

Good answer, Don, but what about Bob's argument?
link to original post

Quote: teliot

Quote: Wizard

Good answer, Don, but what about Bob's argument?
link to original post

Show Spoiler

The definition of summing an infinite series is that its value is equal to the limit of the partial sums. Rearranging the terms creates a different series with different partial sums and a different limit.

In fact you could rearrange the terms of this particular series to converge to any real number you want.

link to original post

I believe him.

Quote: teliot

Quote: Wizard

Good answer, Don, but what about Bob's argument?
link to original post

Show Spoiler

The definition of summing an infinite series is that its value is equal to the limit of the partial sums. Rearranging the terms creates a different series with different partial sums and possibly a different limit.

In fact you could rearrange the terms of this particular series to converge to any real number you want.

You might want to remind yourself of the difference between a convergent series and an absolutely convergent series.

(Edited this to add a few things.)

link to original post

Thanks Eliot. I think we can quit putting discussion in spoiler tags.

I do find it paradoxical we can rearrange the terms in the series and get a different answer. Doesn't one of Peano's Axioms say a+b=b+a?

I'll have to review the difference between a convergent series and an absolutely convergent series.

Maybe I missed it. Was this discussed in the other thread? Did that discussion include a description whether the first operation is addition or subtraction?

I assume not, which leads me to say both Andy and Bob are correct.