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?
Answer
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.
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