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Sum converging to 0by Bipolarity
Tags: converging 
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#1
Nov912, 02:40 PM

P: 783

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 


#2
Nov912, 02:47 PM

P: 350

1+1+.5+5+.25+.25+.125+.125+......



#3
Nov912, 03:30 PM

P: 783

BiP 


#4
Nov912, 03:57 PM

P: 783

Sum converging to 0
Eureka!!! I believe I found it!
[tex] \sum^{\infty}_{n=1} (1)^{n+1} (\frac{1}{2})^{ \frac{2n3+(1)^{n+1}}{4}} [/tex] I believe it converges to 0, but can anyone verify this? BiP 


#5
Nov912, 04:10 PM

P: 685

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
Nov912, 04:15 PM

P: 783

BiP 


#7
Nov912, 08:55 PM

HW Helper
P: 2,263

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
Nov912, 10:18 PM

P: 783

BiP 


#9
Nov912, 11:07 PM

Homework
Sci Advisor
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Thanks
P: 9,645




#10
Nov1012, 05:02 AM

P: 295

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
Nov1012, 07:44 PM

HW Helper
P: 2,263

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 (2k1)\left(\frac{1}{3}\right)^k[/tex] 


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