I have an infinite arithmetic series whose terms are expressed as $$\sum _{k = 1}^{\infty }\frac{2k^2+3}{k} $$ and the sum of it is equal to $$(n + 1)n + 3H_n$$ I can't wrap my head around how they got to this answer.
2026-04-12 17:07:13.1776013633
Why is the $\sum _{k = 1}^{\infty }\frac{2k^2+3}{k} = (n + 1)n + 3H_n$?
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Given the confusion regarding what the question is actually saying (is it going to infinity or to $n$?) and the inclarity based on the formatting of the expression, all I will do is offer a clue.
Try separating the expression in brackets into two separate summations: one of $\dfrac {2 i^2} i = 2i$ and one of $\dfrac 3 i$.
You will find it's a lot simpler than it looked at first.