Question Number 104871 by I want to learn more last updated on 24/Jul/20

Answered by mathmax by abdo last updated on 24/Jul/20

A_p =∫_0 ^1  ((tan^2 x)/x^p )dx ⇒A_p =∫_0 ^1  (((tanx)/x))^2  ×(dx/x^(p−2) )  the function x→(((tanx)/x))^2 ×(1/x^(p−2) ) is continue on]0,1] so integrable  at V(o) f(x) ∼(1/x^(p−2) ) and ∫ (dx/x^(p−2) ) converge ⇔ p−2<1 ⇔p<3

Commented byI want to learn more last updated on 24/Jul/20

Thanks sir. I appreciate.

Commented bymathmax by abdo last updated on 24/Jul/20

you are welcome sir.

Answered by mathmax by abdo last updated on 24/Jul/20

4) let S =Σ_(n=2) ^∞  n x^(n−2)  ⇒S =Σ_(n=0) ^∞ (n+2)x^n  =Σ_(n=0) ^∞  nx^n  +2Σ_(n=0) ^∞  x^n   for ∣x∣<1 we get Σ_(n=1) ^∞  x^n  =(1/(1−x)) ⇒Σ_(n=1) ^∞  nx^(n−1)  =(1/((1−x)^2 )) ⇒  Σ_(n=1) ^∞  nx^n  =(x/((1−x)^2 )) ⇒ S =(x/((1−x)^2 )) +(2/(1−x)) =((x +2(1−x))/((1−x)^2 )) =((x+2−2x)/((1−x)^2 )) ⇒  S =((2−x)/((1−x)^2 )) .