MathLabs

Problem 1

Let Tn=1+2+⋯+n=n(n+1)/2T_n=1+2+\cdots+n=n(n+1)/2 and Sn=1/T1+1/T2+⋯+1/TnS_n=1/T_1+1/T_2+\cdots+1/T_n. Prove that 1/S1+1/S2+⋯+1/S1996>10011/S_1+1/S_2+\cdots+1/S_{1996}>1001.
Step 3 of 5: Reduce to a harmonic sum
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∑n=119961Sn=998+12∑n=119961n\sum_{n=1}^{1996}\frac1{S_n}=998+\frac12\sum_{n=1}^{1996}\frac1n
Detailed analysis

Adding 1/Sn1/S_n for n=1,…,1996n=1,\ldots,1996 gives ∑n=119961/Sn=998+(1/2)∑n=119961/n\sum_{n=1}^{1996}1/S_n=998+(1/2)\sum_{n=1}^{1996}1/n.