Sum of series $1 - 1/3 + 1/6 - 1/10 + ldots$? [duplicate]

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• 
  Sum of an infinite series: $1 + frac12 - frac13- frac14 + frac15 + frac16 - cdots$?
  
  4 answers
  
  

I've been doing a lot of stuff with infinite series and Python lately, and have been getting some very interesting results. For example, the series 1 + 1/16 - 1/256 - 1/4096 + ... (two plus signs, then two minus signs, etc.) converges to 272/257 - wow! But the series 1 + 1/16 + 1/256 + 1/4096 - ... (four alternating plus and minus signs) converges to 69904/65537 - a very interesting result indeed.

I know the series 1 - 1/2 + 1/3 - 1/4 + ... is equal to ln 2. But the series 1 + 1/2 - 1/3 - 1/4 + 1/5 + ... is equal to pi/4 + (1/2 ln 2) - what a strange result!

I also tried a series involving triangular numbers. The series 1 + 1/3 + 1/6 + 1/10 + ... does not seem to converge. What about the series 1 - 1/3 + 1/6 - 1/10 + ... ? There HAS to be some sort of special result to that one, and after 10,000,000 iterations the result stabilized around 0.77258872. Can someone please help me figure this out? (I don't have any experience with calculus)

calculus sequences-and-series

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edited Aug 6 at 18:13

David G. Stork

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asked Aug 6 at 18:10

Anonymous

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Aug 7 at 0:36

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  The alternating series theorem says that any sum of terms that alternate in sign and have magnitudes monotonically decreasing to zero converges. If your series $1+1/3+1/6+1/10+ldots$is $sum_i=1^infty frac 2i(i+1)$ it converges to $2$
  – Ross Millikan
  Aug 6 at 18:16
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  This has already been answered in your previous question. $0.77ldots = 4log 2-2$.
  – Jack D'Aurizio♦
  Aug 6 at 18:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Ok sorry i didnt see it in my other post, someone told me to create a new post
  – Anonymous
  Aug 6 at 18:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Ross Millikan I see now, there was an error in my original program. Thanks!
  – Anonymous
  Aug 6 at 18:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Anonymous Delete this post!
  – Nosrati
  Aug 6 at 18:47
  

  
  
  
  

 | 
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up vote
1
down vote

favorite

This question already has an answer here:

• 
  Sum of an infinite series: $1 + frac12 - frac13- frac14 + frac15 + frac16 - cdots$?
  
  4 answers
  
  

I've been doing a lot of stuff with infinite series and Python lately, and have been getting some very interesting results. For example, the series 1 + 1/16 - 1/256 - 1/4096 + ... (two plus signs, then two minus signs, etc.) converges to 272/257 - wow! But the series 1 + 1/16 + 1/256 + 1/4096 - ... (four alternating plus and minus signs) converges to 69904/65537 - a very interesting result indeed.

I know the series 1 - 1/2 + 1/3 - 1/4 + ... is equal to ln 2. But the series 1 + 1/2 - 1/3 - 1/4 + 1/5 + ... is equal to pi/4 + (1/2 ln 2) - what a strange result!

I also tried a series involving triangular numbers. The series 1 + 1/3 + 1/6 + 1/10 + ... does not seem to converge. What about the series 1 - 1/3 + 1/6 - 1/10 + ... ? There HAS to be some sort of special result to that one, and after 10,000,000 iterations the result stabilized around 0.77258872. Can someone please help me figure this out? (I don't have any experience with calculus)

calculus sequences-and-series

share|cite|improve this question

edited Aug 6 at 18:13

David G. Stork

7,7012929

asked Aug 6 at 18:10

Anonymous

152

marked as duplicate by Nosrati, amWhy calculus
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Aug 7 at 0:36

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  The alternating series theorem says that any sum of terms that alternate in sign and have magnitudes monotonically decreasing to zero converges. If your series $1+1/3+1/6+1/10+ldots$is $sum_i=1^infty frac 2i(i+1)$ it converges to $2$
  – Ross Millikan
  Aug 6 at 18:16
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  This has already been answered in your previous question. $0.77ldots = 4log 2-2$.
  – Jack D'Aurizio♦
  Aug 6 at 18:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Ok sorry i didnt see it in my other post, someone told me to create a new post
  – Anonymous
  Aug 6 at 18:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Ross Millikan I see now, there was an error in my original program. Thanks!
  – Anonymous
  Aug 6 at 18:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Anonymous Delete this post!
  – Nosrati
  Aug 6 at 18:47
  

  
  
  
  

 | 
show 1 more comment

up vote
1
down vote

favorite

up vote
1
down vote

favorite

This question already has an answer here:

• 
  Sum of an infinite series: $1 + frac12 - frac13- frac14 + frac15 + frac16 - cdots$?
  
  4 answers
  
  

I've been doing a lot of stuff with infinite series and Python lately, and have been getting some very interesting results. For example, the series 1 + 1/16 - 1/256 - 1/4096 + ... (two plus signs, then two minus signs, etc.) converges to 272/257 - wow! But the series 1 + 1/16 + 1/256 + 1/4096 - ... (four alternating plus and minus signs) converges to 69904/65537 - a very interesting result indeed.

I know the series 1 - 1/2 + 1/3 - 1/4 + ... is equal to ln 2. But the series 1 + 1/2 - 1/3 - 1/4 + 1/5 + ... is equal to pi/4 + (1/2 ln 2) - what a strange result!

I also tried a series involving triangular numbers. The series 1 + 1/3 + 1/6 + 1/10 + ... does not seem to converge. What about the series 1 - 1/3 + 1/6 - 1/10 + ... ? There HAS to be some sort of special result to that one, and after 10,000,000 iterations the result stabilized around 0.77258872. Can someone please help me figure this out? (I don't have any experience with calculus)

calculus sequences-and-series

share|cite|improve this question

edited Aug 6 at 18:13

David G. Stork

7,7012929

asked Aug 6 at 18:10

Anonymous

152

This question already has an answer here:

• 
  Sum of an infinite series: $1 + frac12 - frac13- frac14 + frac15 + frac16 - cdots$?
  
  4 answers
  
  

I've been doing a lot of stuff with infinite series and Python lately, and have been getting some very interesting results. For example, the series 1 + 1/16 - 1/256 - 1/4096 + ... (two plus signs, then two minus signs, etc.) converges to 272/257 - wow! But the series 1 + 1/16 + 1/256 + 1/4096 - ... (four alternating plus and minus signs) converges to 69904/65537 - a very interesting result indeed.

I know the series 1 - 1/2 + 1/3 - 1/4 + ... is equal to ln 2. But the series 1 + 1/2 - 1/3 - 1/4 + 1/5 + ... is equal to pi/4 + (1/2 ln 2) - what a strange result!

I also tried a series involving triangular numbers. The series 1 + 1/3 + 1/6 + 1/10 + ... does not seem to converge. What about the series 1 - 1/3 + 1/6 - 1/10 + ... ? There HAS to be some sort of special result to that one, and after 10,000,000 iterations the result stabilized around 0.77258872. Can someone please help me figure this out? (I don't have any experience with calculus)

This question already has an answer here:

• 
  Sum of an infinite series: $1 + frac12 - frac13- frac14 + frac15 + frac16 - cdots$?
  
  4 answers
  
  

calculus sequences-and-series

share|cite|improve this question

edited Aug 6 at 18:13

David G. Stork

7,7012929

asked Aug 6 at 18:10

Anonymous

152

share|cite|improve this question

share|cite|improve this question

edited Aug 6 at 18:13

David G. Stork

7,7012929

edited Aug 6 at 18:13

David G. Stork

7,7012929

edited Aug 6 at 18:13

David G. Stork

7,7012929

7,7012929

asked Aug 6 at 18:10

Anonymous

152

asked Aug 6 at 18:10

Anonymous

152

asked Aug 6 at 18:10

Anonymous

152

152

marked as duplicate by Nosrati, amWhy calculus
Users with the  calculus badge can single-handedly close calculus questions as duplicates and reopen them as needed.

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Aug 7 at 0:36

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

marked as duplicate by Nosrati, amWhy calculus
Users with the  calculus badge can single-handedly close calculus questions as duplicates and reopen them as needed.

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Aug 7 at 0:36

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  The alternating series theorem says that any sum of terms that alternate in sign and have magnitudes monotonically decreasing to zero converges. If your series $1+1/3+1/6+1/10+ldots$is $sum_i=1^infty frac 2i(i+1)$ it converges to $2$
  – Ross Millikan
  Aug 6 at 18:16
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  This has already been answered in your previous question. $0.77ldots = 4log 2-2$.
  – Jack D'Aurizio♦
  Aug 6 at 18:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Ok sorry i didnt see it in my other post, someone told me to create a new post
  – Anonymous
  Aug 6 at 18:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Ross Millikan I see now, there was an error in my original program. Thanks!
  – Anonymous
  Aug 6 at 18:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Anonymous Delete this post!
  – Nosrati
  Aug 6 at 18:47
  

  
  
  
  

 | 
show 1 more comment

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  The alternating series theorem says that any sum of terms that alternate in sign and have magnitudes monotonically decreasing to zero converges. If your series $1+1/3+1/6+1/10+ldots$is $sum_i=1^infty frac 2i(i+1)$ it converges to $2$
  – Ross Millikan
  Aug 6 at 18:16
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  This has already been answered in your previous question. $0.77ldots = 4log 2-2$.
  – Jack D'Aurizio♦
  Aug 6 at 18:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Ok sorry i didnt see it in my other post, someone told me to create a new post
  – Anonymous
  Aug 6 at 18:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Ross Millikan I see now, there was an error in my original program. Thanks!
  – Anonymous
  Aug 6 at 18:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @Anonymous Delete this post!
  – Nosrati
  Aug 6 at 18:47
  

  
  
  
  

1

1

The alternating series theorem says that any sum of terms that alternate in sign and have magnitudes monotonically decreasing to zero converges. If your series $1+1/3+1/6+1/10+ldots$is $sum_i=1^infty frac 2i(i+1)$ it converges to $2$
– Ross Millikan
Aug 6 at 18:16

The alternating series theorem says that any sum of terms that alternate in sign and have magnitudes monotonically decreasing to zero converges. If your series $1+1/3+1/6+1/10+ldots$is $sum_i=1^infty frac 2i(i+1)$ it converges to $2$
– Ross Millikan
Aug 6 at 18:16

This has already been answered in your previous question. $0.77ldots = 4log 2-2$.
– Jack D'Aurizio♦
Aug 6 at 18:17

This has already been answered in your previous question. $0.77ldots = 4log 2-2$.
– Jack D'Aurizio♦
Aug 6 at 18:17

Ok sorry i didnt see it in my other post, someone told me to create a new post
– Anonymous
Aug 6 at 18:30

Ok sorry i didnt see it in my other post, someone told me to create a new post
– Anonymous
Aug 6 at 18:30

Ross Millikan I see now, there was an error in my original program. Thanks!
– Anonymous
Aug 6 at 18:32

Ross Millikan I see now, there was an error in my original program. Thanks!
– Anonymous
Aug 6 at 18:32

@Anonymous Delete this post!
– Nosrati
Aug 6 at 18:47

@Anonymous Delete this post!
– Nosrati
Aug 6 at 18:47

 | 
show 1 more comment

2 Answers
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up vote
4
down vote

This is
$$2left(frac11-frac12-frac12+frac13+frac13-frac14-cdotsright)=-2+4ln2.$$

share|cite|improve this answer

answered Aug 6 at 18:12

Lord Shark the Unknown

86k951112

add a comment | 

up vote
1
down vote

Triangular numbers can be given by the formula: $$T_n=fracn(n+1)2$$ Thus, $$sum_n=1^infty (T_n)^-1=2sum_n=1^inftyleft(frac1n-frac1n+1right)=2.$$

Similarly, the sum of alternating triangular numbers can be determined (here you can look at groups of $4$ to handle the alternating sign as well as the partial fraction decomposition).

share|cite|improve this answer

answered Aug 6 at 18:18

Clayton

17.9k22882

add a comment | 

2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
4
down vote

This is
$$2left(frac11-frac12-frac12+frac13+frac13-frac14-cdotsright)=-2+4ln2.$$

share|cite|improve this answer

answered Aug 6 at 18:12

Lord Shark the Unknown

86k951112

add a comment | 

up vote
4
down vote

This is
$$2left(frac11-frac12-frac12+frac13+frac13-frac14-cdotsright)=-2+4ln2.$$

share|cite|improve this answer

answered Aug 6 at 18:12

Lord Shark the Unknown

86k951112

add a comment | 

up vote
4
down vote

up vote
4
down vote

This is
$$2left(frac11-frac12-frac12+frac13+frac13-frac14-cdotsright)=-2+4ln2.$$

share|cite|improve this answer

answered Aug 6 at 18:12

Lord Shark the Unknown

86k951112

This is
$$2left(frac11-frac12-frac12+frac13+frac13-frac14-cdotsright)=-2+4ln2.$$

share|cite|improve this answer

answered Aug 6 at 18:12

Lord Shark the Unknown

86k951112

share|cite|improve this answer

share|cite|improve this answer

answered Aug 6 at 18:12

Lord Shark the Unknown

86k951112

answered Aug 6 at 18:12

Lord Shark the Unknown

86k951112

answered Aug 6 at 18:12

Lord Shark the Unknown

86k951112

86k951112

add a comment | 

add a comment | 

up vote
1
down vote

Triangular numbers can be given by the formula: $$T_n=fracn(n+1)2$$ Thus, $$sum_n=1^infty (T_n)^-1=2sum_n=1^inftyleft(frac1n-frac1n+1right)=2.$$

Similarly, the sum of alternating triangular numbers can be determined (here you can look at groups of $4$ to handle the alternating sign as well as the partial fraction decomposition).

share|cite|improve this answer

answered Aug 6 at 18:18

Clayton

17.9k22882

add a comment | 

up vote
1
down vote

Triangular numbers can be given by the formula: $$T_n=fracn(n+1)2$$ Thus, $$sum_n=1^infty (T_n)^-1=2sum_n=1^inftyleft(frac1n-frac1n+1right)=2.$$

Similarly, the sum of alternating triangular numbers can be determined (here you can look at groups of $4$ to handle the alternating sign as well as the partial fraction decomposition).

share|cite|improve this answer

answered Aug 6 at 18:18

Clayton

17.9k22882

add a comment | 

up vote
1
down vote

up vote
1
down vote

Triangular numbers can be given by the formula: $$T_n=fracn(n+1)2$$ Thus, $$sum_n=1^infty (T_n)^-1=2sum_n=1^inftyleft(frac1n-frac1n+1right)=2.$$

Similarly, the sum of alternating triangular numbers can be determined (here you can look at groups of $4$ to handle the alternating sign as well as the partial fraction decomposition).

share|cite|improve this answer

answered Aug 6 at 18:18

Clayton

17.9k22882

Triangular numbers can be given by the formula: $$T_n=fracn(n+1)2$$ Thus, $$sum_n=1^infty (T_n)^-1=2sum_n=1^inftyleft(frac1n-frac1n+1right)=2.$$

Similarly, the sum of alternating triangular numbers can be determined (here you can look at groups of $4$ to handle the alternating sign as well as the partial fraction decomposition).

share|cite|improve this answer

answered Aug 6 at 18:18

Clayton

17.9k22882

share|cite|improve this answer

share|cite|improve this answer

answered Aug 6 at 18:18

Clayton

17.9k22882

answered Aug 6 at 18:18

Clayton

17.9k22882

answered Aug 6 at 18:18

Clayton

17.9k22882

17.9k22882

add a comment | 

add a comment |