You are here:

Calculus/regarding infirnite series summation


QUESTION: >>>If we take d = 0.4 (or any number less than 1), clearly no matter how large a value of n we choose, the rest of the terms won't be closer to the proposed limit than that.

I could not understand the above point, wherever we stop in the infinite series of 1+1-1+1-1 ... to infinity will either be 1 or 2, why the mathematicians say that it is 0.5.

Also another bizarre claim which mathematicians say infinity is -1/12. How the summation of positive numbers leads to negative number. It simply makes no sense to me. I think we should not perform arithmetic operations (which they do while deriving these bizarre results) should not be done for infinite series.

Please explain.

ANSWER: The series 1+1-1+1-1 is not related to the one you're suggesting at the start.
The theorem to note is that if |d|<1, then the summation from n=0 to n=infinity of d^n
is 1/(1-d).   The series given by 1-1+1-1 is a power series gotten d=-1 and that is not in the range (-1,1), for that range does not include the endpoints.

The bizarre claim about the summation of all positive numbers being 1/12 or the summation of positive numbers being negative - both are not true.  It sounds like those are even taken from a joke on numbers or are the incomplete set.

I use to know a rather elegant proof that came down to 0 = 1.  It was obviously in error somewhere, but the error wasn't seen at first.  Upon close inspection, it could be seen that the equation was divided by 0.  Here it is:

Take n=1.

Clearly, when both sides are multiplied by n, this gives n^2 = n.
When we subtract one from both sides, we get n^2 - 1 = n - 1,
If we factor the left side, we get (n+1)(n-1) = n - 1.
Dividing both sides by n-1 gives n+1 = 1.
Subtract 1 from both sides and get n = 0.
Now we started out assuming that n = 1, but proved n = 0.\

That is from division by 0, for n-1 = 0.

In a similar fashion, the 2nd problem you gave me must be incomplete,
for what was said is false.

---------- FOLLOW-UP ----------

QUESTION: >>It sounds like those are even taken from a joke on numbers or are the incomplete set.

No. It is one of the accepted summation by mathematicians. Please see below:

The address was not found.

The address
stated that if the summation was assumed to have an answer, that answer could be shown to be -1/12, but clearly that summation does not have an answer.  This is because an element n can't be found for an epsilon-delta proof.


All Answers

Answers by Expert:

Ask Experts




Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology (reproduction, insusion of chemicals into bloodstream).


Experience in the area: I have tutored students in all areas of mathematics since 1980. Education/Credentials: BSand MS in Mathematics from Oregon State University, where I completed sophomore course in Physics and Chemistry. I received both degrees with high honors. Awards and Honors: I have passed Actuarial tests 100, 110, and 135.

Maybe not a publication, but I have respond to well oveer 8,500 questions on the PC. Well over 2,000 of them have been in calculus.

I aquired well over 40 hours of upper division courses. This was well over the number that were required. I graduated with honors in both my BS and MS degree from Oregon State University. I was allowed to jump into a few courses at college a year early.

Awards and Honors
I have been nominated as the expert of the month several times. All of my scores right now are at least a 9.8 average (out of 10).

Past/Present Clients
My past clients have been students at OSU, students at the college in South Seattle, referals from a company, friends and aquantenances, people from my church, and people like you from all over the world.

©2017 All rights reserved.