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Enter the license code you found on your CD or download below and get the free Autodesk product.Q: Sum of reciprocals of integers between $1$ and $n$ I need help to solve this exercise. I’m not sure if it’s a legit question. $\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = \frac{n}{n-1}$ Where $n$ is any positive integer. My attempt: First, note that the numerator is an integer. As $n$ increases, the fraction becomes closer to $1$ so the sum becomes closer to $n$. Next, $\frac{1}{n}$ converges to $0$. This implies that if $n$ is small enough, $S$ is a lot larger than $n$. Thus, $\frac{S}{S-1} = \frac{n}{n-1}$. However, I’m unsure about the numerator. I did not use the fact that the numerator is an integer. Can someone help me? A: The denominator should be $(n-1)$ instead of $n-1$. $$ \sum_{k=1}^n \frac{1}{k} = \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\frac{n}{n-1}=\frac{n(n-1)}{n-1}=\frac{n^2-n}{n-1}=n+1-\frac{1}{n} $$ A less formal approach is to notice that the numbers $1,2,\ldots,n$ are equally spaced between $1$ and $n$, so their differences are the same. A: Another way to prove this is to use that the sum is monotonically increasing and hence the sum is always less than $n$. $$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} < \frac{1}{1} + \frac{1}{1} + \

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