Question Number 104406 by Don08q last updated on 21/Jul/20

    Evaluate       (1/2) + (3/8) + (3/(16)) + (5/(64)) + .....

Commented byDon08q last updated on 21/Jul/20

 ((1×2)/4^1 ) + ((2×3)/4^2 ) + ((3×4)/4^3 ) + ((4×5)/4^4 ) + .....

Answered by JDamian last updated on 21/Jul/20

f(x)≡1+x+x^2 +x^3 +x^4 + ∙∙∙ = (1/(1−x))    ∀∣x∣<1  f′′(x)=1∙2+2∙3∙x+3∙4∙x^2 +4∙5∙x^3 +∙∙∙=              =(2/((1−x)^3 ))  f′′((1/4)) = 2 + ((2∙3)/4^1 ) + ((3∙4)/4^2 ) + ((4∙5)/4^3 ) + ((5∙6)/4^4 )+ ∙∙∙ =                 = (2/((1−(1/4))^3 )) = (2/(((3/4))^3 )) = 2((4/3))^3   (1/4)f′′((1/4)) = (1/2)+ ((2∙3)/4^2 ) + ((3∙4)/4^3 ) + ((4∙5)/4^4 ) + ((5∙6)/4^5 )+ ∙∙∙ =                       = (1/2)((4/3))^3

Commented byDon08q last updated on 22/Jul/20

Thank you very much Sir