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Question Number 69374 by mathmax by abdo last updated on 22/Sep/19

let S_n =Σ_(k=1) ^n (((−1)^k )/k) 1) calculate S_n interms of n 2) find lim_(n→+∞) S_n

letSn=∑k=1n(−1)kk1)calculateSnintermsofn2)findlimn→+∞Sn

Commented by mathmax by abdo last updated on 15/Oct/19

1) S_n =W(−1) with w(x)=Σ_(k=1) ^n (x^k /k) w^′ (x) =Σ_(k=1) ^n x^(k−1) =Σ_(k=0) ^(n−1) x^k =((x^n −1)/(x−1)) (we suppose x≠1)⇒ w(x)=∫_0 ^x ((t^n −1)/(t−1))dt +c with c=w(0)=0 ⇒w(x)=∫_0 ^x (t^n /(t−1))dt−∫_0 ^x (dt/(t−1)) =∫_0 ^x (t^n /(t−1))dt −ln∣x−1∣⇒S_n =w(−1)=∫_0 ^(−1) (t^n /(t−1))dt−ln(2) t=−u give ∫_0 ^(−1) (t^n /(t−1))dt =∫_0 ^1 (((−u)^n )/(−u−1))(−du) =(−1)^n ∫_0 ^1 (u^n /(1+u))du ⇒ S_n =(−1)^n ∫_0 ^1 (u^n /(1+u))du−ln(2) 2)we have ∣ S_n +ln(2)∣ =∫_0 ^1 (u^n /(1+u))du≤∫_0 ^1 u^n du =(1/(n+1)) ⇒ lim_(n→+∞) ∣S_n +ln(2)∣=0 ⇒lim_(n→+∞) S_n =−ln(2).

1)Sn=W(−1)withw(x)=∑k=1nxkkw′(x)=∑k=1nxk−1=∑k=0n−1xk=xn−1x−1(wesupposex≠1)⇒w(x)=∫0xtn−1t−1dt+cwithc=w(0)=0⇒w(x)=∫0xtnt−1dt−∫0xdtt−1=∫0xtnt−1dt−ln∣x−1∣⇒Sn=w(−1)=∫0−1tnt−1dt−ln(2)t=−ugive∫0−1tnt−1dt=∫01(−u)n−u−1(−du)=(−1)n∫01un1+udu⇒Sn=(−1)n∫01un1+udu−ln(2)2)wehave∣Sn+ln(2)∣=∫01un1+udu⩽∫01undu=1n+1⇒limn→+∞∣Sn+ln(2)∣=0⇒limn→+∞Sn=−ln(2).

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