Is the sum of all natural numbers $-\frac{1}{12}$? My friend showed me this youtube video in which the speakers present a line of reasoning as to why
$$
 \sum_{n=1}^\infty  n =  -\frac{1}{12}
$$
My reasoning, however, tells me that the previous statement is incorrect:
$$
 \sum_{n=1}^\infty  n = \lim_{k \to \infty} \sum_{n=1}^k n  =  \lim_{k \to \infty}\frac{k(k+1)}{2} = \infty
$$
Furthermore, how can it be that the sum of any set of integers is not an integer. Even more, how can the sum of any set of positive numbers be negative? These two ideas lead me to think of inductive proofs as to why the first statement is incorrect.
Which of the two lines of reasoning is correct and why? Are there any proven applications (i.e. non theoretical) which use the first statement? 
 A: It is a matter of definition. We normally say that a series such as $1-1+1-1+\cdots$ does not converge. It has no value in the limit. If you change the definition of convergence by assigning the value of $1/2$ to that series, then you can expect to get very odd result. Is it useful to do that? Evidently the answer is yes in some applications.
A: [This is a slightly modified version of a previous answer.]
You are right to be suspicious. We usually define an infinite sum by taking the limit of the partial sums. So 
$$1+2+3+4+5+\dots $$
would be what we get as the limit of the partial sums 
$$1$$
$$1+2$$
$$1+2+3$$
and so on. Now, it is clear that these partial sums grow without bound, so traditionally we say that the sum either doesn't exist or is infinite. 
So, to make the claim in your question title, you must adopt a nontraditional method of summation. There are many such methods available, but the one used in this case is Zeta function regularization. That page might be too advanced, but it is good to at least know the name of method under discussion. 
You ask why this nontraditional approach to summation might be useful. The answer is that sometimes this approach gives the correct result in a real world problem. A simple example is the Casimir effect. Suppose we place two metal plates a very short distance apart (in a vacuum, with no gravity, and so on -- we assume idealized conditions). Classical physics predicts they will just be still. However, there is actually a small attractive force between them. This can be explained using quantum physics, and calculation of the magnitude of the force uses the sum you discuss, summed using zeta function regularization. 
A: There are many ways how infinite divergent sums can be manipulated to converge. I suggest you look up Euler summation and Ramanujan summation. One example of this is the sum of the powers of $2$:
$$1+2+4+8+16+...=S$$
If we multiply both sides by two and subtract them, we end up with 
$$S=-1$$
Hence, the sum of infinitely many positive integers can be negative.
We can also "test" our result to see if it works:
$$-1=1+2+4+8+16+...$$
$$0=1+1+2+4+8+16+...$$
$$0=2+2+4+8+16+...$$
$$0=4+4+8+16+...$$
and so on, cancelling out every power of two. 
This convergence property has very important implications in many scientific disciplines, especially physics, where the normalization property is established. This cancels out the infinities involved in quantum theory, string theory, etc. 
A: No.  The terms of of a convergent series must converge to zero.
