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Apply the integral bounds for a finite harmonic sum to the omitted terms from thirteen through one hundred:Adding the given $H_{12}=86021/27720$ gives $5.15338\dots <H_{100}<5.22348\dots$. Both bounds lie between $5.15$ and $5.25$, so every value they permit rounds to the same two significant figures:The rounding follows from the bounds, without computing all the intervening terms.