Regarding the following picture:

Why did the lecturer choose to write O(h^7 f^{(4)}(0)) as the last term in the series? Doesn't it go on forever? And in his notes, he uses O(h^9) instead.

Does it not matter for some reason which term you choose to truncate the series on?

Note that the "MVT" in the last hand-written line stands for Mean Value Thm.

Is that why we can truncate the series?