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 2 of 5: Telescope SnS_n
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Sn=2(1−1n+1)=2nn+1,1Sn=12+12nS_n=2\left(1-\frac1{n+1}\right)=\frac{2n}{n+1},\qquad\frac1{S_n}=\frac12+\frac1{2n}
Detailed analysis

The differences telescope, so Sn=2(1−1/(n+1))=2n/(n+1)S_n=2(1-1/(n+1))=2n/(n+1) and 1/Sn=1/2+1/(2n)1/S_n=1/2+1/(2n).