โจทย์เลขสมัยนี้ Σ(-1)ⁿ = ? ; for n=0,... ∞

This week-end, I like to show you a ‘mathematical object’ (examples of math objects are numbers/scalars, vectors, tensors, functions, graphs, nets, trees,…), some ‘words’ in math language and some ideas about ‘talks’ about ‘math’. I hope I can persuade you to look at mathematics differently 🙊️.

First, let us see what  means.  mathematics  n : a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement (nickname: math,  maths); in Thai คณิตศาสตร์ [คะนิดตะสาด] น. วิชาว่าด้วยการคำนวณ. ( ส. คณิต + ศาสฺตฺร). คณิต: น. การนับ, การคำนวณ. คำนวณ: ก. กะประมาณ, คิดหาผลลัพธ์โดยวิธีเลข. As you can see, math is basically  about using logical/reasonable process to search for solutions (of problems) from ‘numbers’ or measurements.

We know what numbers are (I will assume that we have learned to count – that to measure how many objects of interest there are). We use Arabic alphabet/symbols (0,1,2,3,…) to write about numbers. (Words for these numbers are of course different in different languages. English has been officially adopted as the language of mathematics, as well as of Science, Engineering, Space,…). I will also assume that we have learned to count and learned that ‘+ ’ symbol (read it as ‘plus’)  means ‘add’ (put more objects of the same 'type' (set, class, category,…) or ‘interest’ into a ‘container’ (the same heap or pile or box or place or ,,, ). We call the result of adding - the 'sum of'. And we write the adding in math language as  Σ(a₁  a₂  a₃  a₄, …, aₙ) ; for n=1,... (read it as ‘the sum of [a sequence] a n for n equal 1 to infinity’). ‘a’ is the symbol we choose to represent a number, and we use subscript to index or tell which number in the list. We use ‘(..)’ symbol to ‘contain’ the numbers in the list. ‘=’ symbol means ‘equal’ (or ‘equals’). Σ is called ‘summation’ or 'sum of'. We normally use superscript to mean ‘to the power of’ so aⁿ means a to the power n or a multiplies itself n times. 

 We have learned about ‘-’ (minus - read it ‘my ners’ or ‘sub tract’ or ‘take away’) and ‘negative numbers’ like -1, -2, -3, -n,.. or -n. And a simple rule in math: a + -b equal a - b

What is this infinity(∞)? In counting, we mean a number so much more than any number we can think of. (We can think of a lot and add some more to it and we will not get to infinity yet because infinity is ‘unreachable’. If we start count from a number (say 1), we go through 1,2,3,4,..,n,n+1,n+2,… until we have enough counting. We abbreviate the sequence of numbers, in counting, as ‘…’ (exactly 3 dots) or as ∞ (a sleeping number 8 ;-). What is a ‘sequence’ (in math language)? By a ‘sequence’ (often write as S - capital S), we mean a list (or  parade) of (math objects or) numbers, sometimes arranged to show a certain distinct quality (property, attribute, characteristic). So we see Math language is not really ‘exact’ but ‘shorthand’ right? To be good in math, we have to learn this shorthand way of saying things. 

Now we are ready to tackle this math puzzle. It goes like this: we have a sequence (1,-1,1,-1,…) --that is a list of infinitely many alternating positive and negative numbers. The question is ‘how’ do we find the sum of these numbers in the list. We are to learn how to work with math (reasoning). We are not really interested in remembering answers like 1 or 0. We want to learn/know how we arrive at the answer.

Let me show an example of working out a solution: Σ (-1)ⁿ  ;  for n=0,... ∞ ?

By expanding math language  Σ (-1)ⁿ ; for n = 0,1,2,… (Grandi's series) we get

1-1+1-1+1-1+1-1+1-1+1-...

Let us try adding 2 numbers at a time; so we get 

(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)-... = 0 if n is even|S ends with -1, 1 if n is odd|S ends with 1 ---answer 1

But if we start from the second number, we get

1-(1+1)-(1+1)-(1+1)-(1+1)-(1+1)-(1+1)+... = 1-2-2-2-2-2... = ?  if infinity is odd;  ? if ∞ is even --answer 2

What if we start from the third number 

1-1+(1-1)+(1-1)+(1-1)+...  = ?

What happens if we try adding 2 numbers and skip a number in the list that is 

(1-1)+1-(1+1)-1+(1-1)+1-(1+1)-1+(1-1)... = 0+1+2 -1-0+1-2+0-... = ?

What if we try adding 3 number at a time

(1-1+1)-(1+1-1)+(1-1+1)-(1+1-1)-... = 1 - 1 + 1 - 1… --Ummh it looks like we are back to the start 

 Or if we start from the second number
1-(1+1-1)+(1-1+1)-(1+1-1)+(1-1+1)+... = 1 - 1 + 1 - 1 + 1 …

OK. I We will need some time to work this out. I will come back in a few days to add more food for math.

[Edited and added on 29 May 2565:

Other ways to do the sum:

Let   S = 1 − 1 + 1 − 1 + ..., so

If last number is -1:  S = 1 − 1 + 1 − 1 + ... -1; so 1 - S = 1 - (1 − 1 + 1 − 1 + ... -1) = 1 - 1 + 1 - 1 + 1...+1 = S + 1; or 1- S = S +1 or 0 = 2S which gives S = 0

if last number is +1;  S = 1 − 1 + 1 − 1 + ... +1; so 1 - S = 1 - (1 − 1 + 1 − 1 + ... +1) = 1 - 1 + 1 - 1 + 1...-1 = S - 1; or 1- S = S - 1 or 2 = 2S which gives S = 1

If we do not know what the last number is  S = 1 − 1 + 1 − 1 + …
   1 − S = 1 − (1 − 1 + 1 − 1 + ...) = 1 − 1 + 1 − 1 + ... = S
   1 − S = S and 1 = 2S --> S = 1/2 --answer 3 *note this is based on ‘the definition of infinity’| 1+ ∞ =  ∞

Infinity is the cause to many different paradoxes in mathematics. This is also how Srinivasa Ramanujan discovered that the sum of all natural numbers to infinity is -1/12 (see https://en.wikipedia.org/wiki/Ramanujan_summation ). 

** Infinity ∞ is a useful concept but it is not a number (belonging to neither integer Z nor real R). Therefore math operations on ∞ (like ∞+a, ∞*a and log∞ or √∞ ) are undefined (meaning ‘have no meaning’ ;-).

By the way, look up https://en.wikipedia.org/wiki/Grandi%27s_series if you can't wait. ;-)