Does Mathematical Induction Hold for Infinite Sums?

[wany]

Ok so I have a question regarding induction.

So suppose that a sum from 1 to n=1 has property P. Suppose further that if the sum from 1 to n has property P, then the sum from 1 to n+1 has property P also (for n greater than or equal to 1). Now will this property hold for the sum from 1 to infinity?

I know that from mathematical induction this should be true, but I am wondering does it carry out to infinity. There's just something in the back of my head telling me that for some reason this might not hold.

So I guess, I just want to get some clarification.

 

[JCVD]

It does not necessarily carry out to infinity; induction only tells you your proposition holds for all natural numbers n.

Consider the proposition P(n) that the sum 1+2+...+n is finite. P(1) is true, and if P(n) holds, then 1+2+...+n is finite, so 1+2+...+n+(n+1) is also finite, so P(n+1) holds. Thus P(n) is true for all n (not that induction is necessary here). However, if we formulate P(infinity) to be the proposition that 1+2+3+... is finite, then P(infinity) is false.

 

[robert Ihnot]

If induction goes over the entire set of natural numbers then you can call this the infinite case. Which is the same as for every natural number, i.e., we have a sucessor--namely n passes to n+1. Thus we may conclude that all natural numbers have been included in the set and such a set is infinite, since we have no largest number.

In some cases, this is called the Principal of Induction and otherwise it is called the Axiom of Induction. Usually it is considered an axiom, so that no proof is necessary. However in some systems by defining the number system, well-ordering, and the successor function, it can be proven.

Principally, it is probably better at the start to simply regard it as an axiom. That is the simplist way to see it. There are different forms of infinity as Cantor has shown, but what occurs here is the simplist case: countable infinity. That is, we can come up--at least mentally--with a long list and given any number we can find it somewhere on this list.

 

[ramsey2879]

Ok so I have a question regarding induction.

So suppose that a sum from 1 to n=1 has property P. Suppose further that if the sum from 1 to n has property P, then the sum from 1 to n+1 has property P also (for n greater than or equal to 1). Now will this property hold for the sum from 1 to infinity?

I know that from mathematical induction this should be true, but I am wondering does it carry out to infinity. There's just something in the back of my head telling me that for some reason this might not hold.

So I guess, I just want to get some clarification.

I assume you proved the fact that the property holds for n = 1; and, also that if holds for a given integer n, that it must hold for n + 1. If so, then the property holds for all integers > 0 no matter how large. That is the power of induction.

 

[wany]

Hmmm I see. So in the case of JCVD's comment, P(infinity) would actually still follow the property that P(k) is finite?

 

[disregardthat]

Induction is just a proof of a general statement P(n) for each n. From the inductive argument, you may, for any chosen integer m, infer
P(m). P(infinity) does not make sense. Induction does not say anything about the "infinite" case, if it even makes sense in some informal way.

However, consider the following example. Say you prove by induction that 1/2+1/2^2...+1/2^n = 1-1/2^n. You can from this deduce the value of the corresponding infinite sum (the limit of the sequence of partial finite sums), but this is not the "infinite case" in the induction argument. The value of the limit is not proven in the induction argument, but deduced on that basis.

 

[wany]

I see, well thank you very everyone's help.