# Harmonic Series sums to One?

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: and this was the solution: I just don't get this part: I deeply apologize if its something trivial. Its been a while since my last math class in linear algebra. Any hint would be appriciated!

• 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? – Peter Foreman Aug 30 '19 at 18:38
• 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. – Kenneth Dang Aug 30 '19 at 18:45

$$\sum_{j=1}^m\frac1m\ne\sum_{j=1}^m\frac1j.$$
$$\sum_{j=1}^m\frac1m=\overbrace{\frac1m+\frac1m+\cdots+\frac1m}^{m\text{ times}}=1$$