do you know any other weird convergent series?

[DuctTapePro]

hi, i only know 4 weird convergent series. where can i find the other? do you know any source? or at least do you know other than
1+2+3+4+5+6+7+8... = -1/12
1-1+1-1+1-1+1-1... = 1/2
1-2+3-4+5-6+7-8... = 1/4
1+2+4+8+16+32+64... = -1

 

[Dr.Peterson]

hi, i only know 4 weird convergent series. where can i find the other? do you know any source? or at least do you know other than
1+2+3+4+5+6+7+8... = -1/12
1-1+1-1+1-1+1-1... = 1/2
1-2+3-4+5-6+7-8... = 1/4
1+2+4+8+16+32+64... = -1

These aren't convergent series; they are divergent series for which some alternative "summation method" can provide a finite value.

For examples of such methods, and series they work on, see https://en.wikipedia.org/wiki/Divergent_series . Here are some examples listed there:

• 1 - 1 + 1 - 1 + · · · "
  
  "
  
• 1 − 2 + 3 − 4 + · · · "
  
  "
  
• 1 − 1 + 2 − 6 + 24 − 120 + ... "
  
  "
  
• 1 − 2 + 4 − 8 + ⋯ "
  
  "
  
• 1 + 2 + 4 + 8 + ⋯ "
  
  "
  
• 1 + 1 + 1 + 1 + · · · "
  
  "
  
• 1 + 2 + 3 + 4 + · · · "
  
  "
  

Notice the "scare quotes" around the equal signs -- they mean that the numbers on the right are not actually equal to the series, but are associated with them by some means that makes some sense in some context.

 

[topsquark]

1 − 1 + 2 − 6 + 24 − 120 + ... "

"

Thank you. I have never seen this one.

-Dan

 

[mmm4444bot]

… 1-1+1-1+1-1+1-1... = 1/2 …

Using the Associative Property, this infinite series also "equals" both 0 and 1.

(1 - 1) + (1 - 1) + (1 - 1) + (1 - 1) + … \(\displaystyle \;\) = \(\displaystyle \;\) 0 + 0 + 0 + 0 + …

1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + … \(\displaystyle \;\) = \(\displaystyle \;\) 1 + 0 + 0 + 0 + …


The fact that we can manipulate summation techniques to obtain different sums {0, 1/2, 1} shows that something is wrong with our logic. The error in logic is often assuming a series is convergent (at the beginning) when in fact it is not.

Before we attempt to sum an infinite series, we must examine the sequence of partial sums. If these partial sums approach a finite value (i.e., the limit exists) or they approach ±∞, then the sum of the infinite series is the limit (or, in the case of ±∞, the sum is infinite in magnitude).

The partial sums for the infinite series above are: 1, 0, 1, 0, 1, 0, …

Clearly, the partial sums are not approaching any finite value (or ±∞). Hence, the series has no sum. :cool:

 

[mmm4444bot]

… 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 ... = -1/12

1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 ... = 1/2

This is false, and that is based on this ! :cool: