I am supposed to prove that $(1/1) + (1/3) + (1/6) + \dots + (1/t_n) < 2$.
The hint is that $$ \frac2{n(n+1)} = 2\left(\frac1{n} - \frac1{n+1}\right) $$
However, I was thinking that if you take $ 2\left(\frac{1}{t_n} + \frac{1}{t_{n+1}}\right) = \frac1{t_{\frac{n}2}}$ where $n$ is even.
So my logic is you can subtract 1 from both sides multiply by 2 and you are back where you started.
eg $$ 1-1 + \frac{1}{3} + \frac{1}{6} + \frac{1}{10} + \frac{1}{15} + ... < 2 -1$$ $$ 2\left(\frac{1}{2} + \frac{1}{6} + ... \right) < 1 \times 2 $$ $$ 1 - 1 + \frac{1}{3} + ... < 2 -1$$ etc
Is this correct logic or am I missing something? Obviously by the hint this was not the proof they had in mind. So how would I prove this using the hint?