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Question Number 211594 by mnjuly1970 last updated on 14/Sep/24

If , H_n ^( −) =1−(1/2) +(1/3) −...+(((−1)^(n+1) )/n) prove that:Σ_(n=1) ^∞ ((H_n ^( − ) −ln(2))/n)=ln^2 (2) −−−−−−−−−−

If,H−n=1−12+13−...+(−1)n+1nprovethat:∑∞n=1H−n−ln(2)n=ln2(2)−−−−−−−−−−

Commented by Ghisom last updated on 14/Sep/24

just a hint: H_n ^� =Σ_(j=1) ^n (((−1)^(j+1) )/j) ⇒ lim_(n→∞) H_n ^� =ln 2 [Taylor series of ln (x+1); ∣x∣<1∨x=1]

justahint:H¯n=∑nj=1(−1)j+1j⇒limn→∞H¯n=ln2[Taylorseriesofln(x+1);∣x∣<1∨x=1]

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