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I apologise for the inefficient typesetting of the question, this is my first time on the website. I was thinking about two series that are defined as below:

S1 = {-2, -3, -4, -5, -6 ... -∞} to negative infinity

S2 = {+6, +7, +8 +9, +10 ... ∞} to positive infinity.

If one were to take the sum of two series, there seems to be 2 different values for the some. One is a finite value, you can observe that in the first series given after the value of "-6" every value of same magnitude but different sign can be found (-6 and +6, -7 and +7 and so on...) this leaves the sum with only different values being -2-3-4-5 which equals to -14.

On the other hand we could sum the same indice of each given series such as:

(-2+6) + (-3+7) + (-4+8) ... to infinity.

On the second sum, each pair of indice equals to 4 and since there are infinite integers the sum would be 4 * ∞ = ∞.

We could create a generalised rule for the sum of such two finite series but in this example they both go to infinity. My question now is, which one is the true answer to our question? How could we properly represent the summation of such two series?

itagaji
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    This is hard to follow. First of all, do you mean "series"? A series is a sum, $a_1+a_2+\cdots$. Your $S_1, S_2$ look like sequences, not series. Secondly, how are you defining the sum here? Neither of these series (if you really meant series) would converge. – lulu Jun 13 '23 at 14:26
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    If you are asking about divergent series, the plain fact is that it's easy to get apparently different sums. $1-1+1-1+\cdots=0$ since it is $(1-1)+(1-1)+\cdots$. But it is also $1$ since you can write it as $1-(1-1)-(1-1)-\cdots$. You just can't manipulate divergent series this way. – lulu Jun 13 '23 at 14:28
  • Your question is very broad, and to answer it would basically require one to lay out all definitions that are used in calculus when analyzing series. Take any mathematical textbook, first class calculus, and read the chapter on series. All your questions have answers there, but those answers are too long for the form factor this site is meant for. – 5xum Jun 13 '23 at 14:33
  • For example, a good intro to series can also be found here. https://mathworld.wolfram.com/Series.html You should be to see that just by definitions provided in my link, it's obvious that the two series you cite do not converge (and therefore do not have a sum). – 5xum Jun 13 '23 at 14:35
  • @5xum what if I wanted to find out the sum of all the integer values that the variable "a" can have for the following inequality: [a^2 - 4a + 5 > 0] here we face again the similar kind of situation where a can get all integer values smaller than -1 and bigger than 5. How would you find the answer? – itagaji Jun 13 '23 at 14:41
  • I would say that the question is nonsensical and does not need to have an answer. – JMoravitz Jun 13 '23 at 14:42
  • @JMoravitz Sure, but that's because OP does not understand the basics, so I don't see anything wrong in pointing him towards the basics... – 5xum Jun 13 '23 at 14:43
  • @5xum and the others who took the time to answer to my question, I would like to start of by thanking you. The inequality and the question above was an actual problem that was represented in one of my textbooks. The book claimed that the answer was -14 because after -6 all the values exist for both negative and positive infinity of roots. I found it weird and thought about it and found that it can be equal to a lot of weird things when it goes to infinity. So the question was nonsensical after all. – itagaji Jun 13 '23 at 14:47
  • @itagaji Well, yes, the sum of all integers for which $a^2-4a + 5>0$ is "a sum of an infinite set". Unlike finite sets, infinite sets are much trickier to define "sums" on. In particular, there is no "natural" way to define a sum. For finite sets, because addition is associative and commutative, there exists a single possible consistent definition of what a sum of a set is. This is not so with infinite sets. This is why the concept of a series was invented by mathematitians. – 5xum Jun 13 '23 at 14:47
  • @itagaji Can you list the exact book you got the problem from? – 5xum Jun 13 '23 at 14:48
  • @5xum the book is in my native language which you probably would not understand. but it basically asked for the sum of all the possible integer values for the "a" variable in the inequality. i guess the author did not pay attention. – itagaji Jun 13 '23 at 14:52
  • @itagaji Depends on the book and the author, I guess :) – 5xum Jun 13 '23 at 14:53
  • Closely related: Intuitively, why does $\int\limits_{-\infty}^\infty \sin(x)dx$ diverge?. Note there the distinction between $\lim\limits_{n\to\infty}\int\limits_{-n}^n f(x)dx$ and $\lim\limits_{a\to-\infty,b\to\infty}\int\limits_a^bf(x)dx$, and how that relates to your "begin the summation centered using opposing terms adding to 4 vs centered with opposing terms adding to 0." – JMoravitz Jun 13 '23 at 15:01

1 Answers1

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The series that you showed $$ -6+6-7+7-8+8-\ldots $$ does not converge, as you will be oscillating around $0$ like $$ -6, 0, -7, 0, \ldots $$ However, as you have seen you can get your sum to do a lot of bizarre things by changing the order in which you add terms together. This is because infinite sums are not guaranteed to keep the same value when you shuffle them around. For conditionally convergent sums this would be the Riemann Rearrangement Theorem

wjmccann
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