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Let-s-n-c-1-1-1-2-n-c-2-1-1-2-1-3-n-c-3-1-n-1-1-1-2-1-3-1-n-n-c-n-then-prove-that-s-n-1-

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Question Number 29196 by mathshooter last updated on 05/Feb/18
Let s = n_c_1 − (1+(1/2))n_c_2 +(1+(1/2)+(1/3))n_c_3 +.......+(−1)^(n−1) (1+(1/2)+(1/3)+....+(1/n))n_c_n then prove that s×n =1.
$$\mathrm{Let}\:\mathrm{s}\:=\:\mathrm{n}_{\mathrm{c}_{\mathrm{1}} } \:−\:\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}\right)\mathrm{n}_{\mathrm{c}_{\mathrm{2}} } \:+\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}\right)\mathrm{n}_{\mathrm{c}_{\mathrm{3}} } \\ $$$$+…….+\left(−\mathrm{1}\right)^{\mathrm{n}−\mathrm{1}} \left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+….+\frac{\mathrm{1}}{\mathrm{n}}\right)\mathrm{n}_{\mathrm{c}_{\mathrm{n}} } \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{s}×\mathrm{n}\:=\mathrm{1}. \\ $$

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