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I don't get it. How does $\sum_{j=1}^m\frac{1}{m}=1$? This looks like a harmonic series. I got this from brilliant.org, the website that trains students for AMC, AIME, Olympiad type of problems. This was the original problem: enter image description here and this was the solution: enter image description here
I just don't get this part: enter image description here I deeply apologize if its something trivial. Its been a while since my last math class in linear algebra. Any hint would be appriciated!

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    $\begingroup$ What are the values of the sums$$\sum_{j=1}^5\frac15\quad\sum_{j=1}^{97}\frac1{97}$$for example? Why is using the letter $m$ any different? $\endgroup$ – Peter Foreman Aug 30 '19 at 18:38
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    $\begingroup$ I get it now, it was a little confusing. I thought the sum $\sum_{j=1}^5 \frac{1}{m}$ would be $\frac{1}{1}+\frac{1}{2]+\frac{1}{3}+...+\frac{1}{5}$ that's why. It's actually that the bottom is constant. I'm such a dummy, I'll accept the answer below soon. Thanks so much for helping though. $\endgroup$ – Kenneth Dang Aug 30 '19 at 18:45
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$$\sum_{j=1}^m\frac1m\ne\sum_{j=1}^m\frac1j.$$

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  • $\begingroup$ Thanks, this was my confusion. I'll accept your answer in 2 minutes, when I can. $\endgroup$ – Kenneth Dang Aug 30 '19 at 18:49
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$$\sum_{j=1}^m\frac1m=\overbrace{\frac1m+\frac1m+\cdots+\frac1m}^{m\text{ times}}=1$$

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