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Question Number 136404 by mathmax by abdo last updated on 21/Mar/21
calculate Σ_(n=0) ^∞ ((n(−1)^n )/((2n+1)^2 (n+3)))
calculate∑n=0∞n(−1)n(2n+1)2(n+3)
Answered by mindispower last updated on 21/Mar/21
−(1/2){Σ_(n≥0) (((−1)^n )/((2n+1)(n+3)))−Σ_(n≥0) (((−1)^n )/((2n+1)^2 ))}=−(1/2)A A=Σ(2/5).(((−1)^n )/(2n+1))−(1/5)Σ_(n≥0) (((−1)^n )/(n+3)) =(2/5)Σ_(n≥0) (((−1)^n )/(2n+1))−(1/5)Σ(((−1)^(n+3) )/(n+3)) =(2/5)tan^(−1) (1)−(1/5)Σ_(n≥2) (((−1)^n )/(n+1))−G =(π/(10))−(1/5)(ln(2)−(1/2))−G we get −(π/(20))+((ln(2))/(10))−(1/(20))+(G/2),G catalan constante
−12{∑n⩾0(−1)n(2n+1)(n+3)−∑n⩾0(−1)n(2n+1)2}=−12AA=Σ25.(−1)n2n+1−15∑n⩾0(−1)nn+3=25∑n⩾0(−1)n2n+1−15Σ(−1)n+3n+3=25tan−1(1)−15∑n⩾2(−1)nn+1−G=π10−15(ln(2)−12)−Gweget−π20+ln(2)10−120+G2,Gcatalanconstante

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