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Question Number 145951 by ArielVyny last updated on 09/Jul/21

Σ_(n≥1) (((−1)^n )/n)=??

∑n⩾1(−1)nn=??

Answered by Olaf_Thorendsen last updated on 09/Jul/21

(1/(1+x)) = 1−x+x^2 −x^3 +... = Σ_(n=0) ^∞ (−1)^n x^n ln∣1+x∣ =Σ_(n=0) ^∞ (−1)^n (x^(n+1) /(n+1)) = −Σ_(n=1) ^∞ (−1)^n (x^n /n) if x = 1 Σ_(n=1) ^∞ (((−1)^n )/n) = −ln2

11+x=1−x+x2−x3+...=∑∞n=0(−1)nxnln∣1+x∣=∑∞n=0(−1)nxn+1n+1=−∑∞n=1(−1)nxnnifx=1∑∞n=1(−1)nn=−ln2

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