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Question-84835




Question Number 84835 by Power last updated on 16/Mar/20
Commented by abdomathmax last updated on 16/Mar/20
S=Σ_(k=1) ^(32)  (k/(k+1)) =Σ_(k=1) ^(32)  ((k+1−1)/(k+1))  =Σ_(k=1) ^(32) (1)−Σ_(k=1) ^(32)  (1/(k+1))  =32−Σ_(k=2) ^(33)  (1/k) =32−Σ_(k=1) ^(33)  (1/k)+1  =34−H_(33)    with H_n =Σ_(k=1) ^n  (1/k)
$${S}=\sum_{{k}=\mathrm{1}} ^{\mathrm{32}} \:\frac{{k}}{{k}+\mathrm{1}}\:=\sum_{{k}=\mathrm{1}} ^{\mathrm{32}} \:\frac{{k}+\mathrm{1}−\mathrm{1}}{{k}+\mathrm{1}} \\ $$$$=\sum_{{k}=\mathrm{1}} ^{\mathrm{32}} \left(\mathrm{1}\right)−\sum_{{k}=\mathrm{1}} ^{\mathrm{32}} \:\frac{\mathrm{1}}{{k}+\mathrm{1}} \\ $$$$=\mathrm{32}−\sum_{{k}=\mathrm{2}} ^{\mathrm{33}} \:\frac{\mathrm{1}}{{k}}\:=\mathrm{32}−\sum_{{k}=\mathrm{1}} ^{\mathrm{33}} \:\frac{\mathrm{1}}{{k}}+\mathrm{1} \\ $$$$=\mathrm{34}−{H}_{\mathrm{33}} \:\:\:{with}\:{H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$

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