TITLE: Is it possible to convert a divergent series by subtracting a constant?
QUESTION [5 upvotes]: This question came to my mind after learning about the existence of the Euler Mascheroni constant. I think that if each term of the divergent series is depressed by a certain amount then maybe the series might become convergent. However, this is just my fancy and if there is solid argument that proves that this is impossible then I would greatly appreciate as I was planning to do research on this question.

REPLY [3 votes]: Not always. Consider $1+2+3+4+\dotsb$; this diverges no matter what constant you subtract from the terms. That is:
$$(1-c)+(2-c)+(3-c)+(4-c)+\dotsb$$
always diverges.