Can you construct a sequence of real nonzero numbers whose sum converges to 0?

In summary, the author does not think that a sequence of real nonzero numbers whose sum converges to 0 exists. Some people argue that this sequence can be constructed, but it does not converge to 0.
  • #1
Bipolarity
776
2
Does there exist a sequence of real nonzero numbers whose sum converges to 0?
I would think there isn't, but I'm interested in people's opinions and arguments.

For any nonzero m, a series of nonzero numbers whose sum converges to m can easily be constructed using the formula: [itex] \sum ^{\infty}_{n=1}m(0.5)^{n} [/itex]

But that is for nonzero m, what if you wanted to construct a series whose sum converged to 0?

BiP
 
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  • #2
1+-1+.5+-5+.25+-.25+.125+-.125+...
 
  • #3
Vargo said:
1+-1+.5+-5+.25+-.25+.125+-.125+...

Can you find an explicit representation for that seqence (i.e. with sigma notation) ?

BiP
 
  • #4
Eureka! I believe I found it!

[tex] \sum^{\infty}_{n=1} (-1)^{n+1} (\frac{1}{2})^{ \frac{2n-3+(-1)^{n+1}}{4}} [/tex]

I believe it converges to 0, but can anyone verify this?

BiP
 
  • #5
If [itex]\sum ^{\infty}_{n=1}m(0.5)^{n}[/itex] converges to m, then shouldn't [itex]m-\sum ^{\infty}_{n=1}m(0.5)^{n}[/itex] converge to 0?
 
  • #6
Edgardo said:
If [itex]\sum ^{\infty}_{n=1}m(0.5)^{n}[/itex] converges to m, then shouldn't [itex]m-\sum ^{\infty}_{n=1}m(0.5)^{n}[/itex] converge to 0?

Yes, but [itex]m-\sum ^{\infty}_{n=1}m(0.5)^{n}[/itex] is not a series... unless you can express it as one with nonzero terms.

BiP
 
  • #7
Who cares about expressing it as one with nonzero terms? A series is a series is a series.
This is simple
How about
[tex]\sum_{n=0}^\infty \frac{(\pi)^{2n+1}}{(2n+1)!} (-1)^n[/tex]
 
  • #8
lurflurf said:
Who cares about expressing it as one with nonzero terms? A series is a series is a series.

The problem requires it.

BiP
 
  • #9
Bipolarity said:
Yes, but [itex]m-\sum ^{\infty}_{n=1}m(0.5)^{n}[/itex] is not a series... unless you can express it as one with nonzero terms.
How about (1-λ)λn-(1-μ)μn for n >= 0, where λ is algebraic and μ is not?
 
  • #10
How about taking a sequence [itex](a_x)_x[/itex] which satisfies [itex]\displaystyle \lim_{x\to\infty}a_x = 0[/itex] and then using the series [itex]\displaystyle \sum_{x=0}^{\infty} (-1)^x b_x[/itex], where the sequence [itex]b_x[/itex] is defined as [itex]b_{2x} = b_{2x+1} = a_x[/itex]?
 
  • #11
So you dislike the pi example and the usual example
[tex]\sum_{k=0}^\infty a_k b_x[/tex]
where a_k is a sequence of positive numbers tending to zero and B_k is any sequence of -1 and 1 such that the series tends to zero.
What about any number of obvious examples such as
[tex]\sum_{k=0}^\infty (2k-1)\left(\frac{1}{3}\right)^k[/tex]
 
Last edited:

1. What does it mean for a sum to converge to 0?

When a sum converges to 0, it means that the terms in the sum are getting smaller and smaller, eventually approaching 0 as the number of terms increases. This indicates that the sum is approaching a finite value, in this case 0.

2. How do you determine if a sum converges to 0?

To determine if a sum converges to 0, you can use various mathematical tests such as the Ratio Test or the Comparison Test. These tests evaluate the behavior of the sum as the number of terms increases and determine if the sum approaches a finite value, in this case 0.

3. Can a sum converge to 0 if the terms are all positive?

Yes, a sum can still converge to 0 even if all the terms are positive. This is because the terms can still approach 0 as the number of terms increases, even if they are always positive.

4. What happens if a sum does not converge to 0?

If a sum does not converge to 0, it means that the terms in the sum do not approach a finite value as the number of terms increases. This can indicate that the sum is either divergent or oscillatory, meaning it alternates between positive and negative values without approaching 0.

5. Is it possible for a sum to converge to 0 but not have a finite number of terms?

Yes, it is possible for a sum to converge to 0 without having a finite number of terms. This is known as an infinite sum or series, where the sum continues indefinitely but approaches a finite value of 0. Examples of this include the harmonic series or the geometric series with a common ratio of less than 1.

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