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 1 of 5: Rewrite each reciprocal triangular number
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1Tm=2m(m+1)=2(1m−1m+1)\frac1{T_m}=\frac2{m(m+1)}=2\left(\frac1m-\frac1{m+1}\right)
Detailed analysis

Since Tm=m(m+1)/2T_m=m(m+1)/2, factorization gives 1/Tm=2(1/m−1/(m+1))1/T_m=2(1/m-1/(m+1)).