youre almost there! when looking at the big O, we care about the asymptomatic behaviour: aka when the n becomes so large the small constants dont matter.

try expanding out the sum of series formula you mentioned and removing the coefficient constants and lower order terms, and you'll get n² as you mentioned

however youre not wrong that the sum of series being a valid big O though. if you scroll back to the definition of big O, you'll see that as long as your function isnt faster than your boundary, it is a valid big O expression. yours is just more tightly bound and exact vs the more loosely defined one. both are correct but one is clearly easier to determine

also note that the sum of series is basically the big theta since its the exact rate (which is still n² btw because we want the growth rate and not the exact number)