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 5 of 5: Finish the inequality
Previous stepNext step
998+12∑n=119961n>998+62=1001998+\frac12\sum_{n=1}^{1996}\frac1n>998+\frac62=1001
Detailed analysis

Substitution yields 998+(1/2)∑n=119961/n>998+6/2=1001998+(1/2)\sum_{n=1}^{1996}1/n>998+6/2=1001.