Question Number 101342 by mathmax by abdo last updated on 02/Jul/20

find  U_n =∫_0 ^1  ((x^(2n) −1)/(lnx))dx  with n integr natural and n≥2  find nature of the serie Σ U_n

Answered by mathmax by abdo last updated on 04/Jul/20

U_n =∫_0 ^1  ((x^(2n) −1)/(lnx))dx changement lnx =−t give x =e^(−t)   U_n =−∫_0 ^∞   ((e^(−2nt) −1)/(−t))(−e^(−t) )dt =−∫_0 ^∞  ((e^(−(2n+1)t) −e^(−t) )/t)dt  =∫_0 ^∞  ((e^(−t) −e^(−(2n+1)t) )/t)dt  let f(x) =∫_0 ^∞  ((e^(−t) −e^(−(2n+1)t) )/t)e^(−xt)  dt  withx>0 ⇒  f^′ (x) =−∫_0 ^∞   (e^(−(1+x)t) −e^(−(2n+1+x)t) )dt  =∫_0 ^∞ (e^(−(2n+1+x)t) −e^(−(1+x)t) )dt =[((−1)/((2n+1+x)))e^(−(2n+1+x)t) +(1/(1+x))e^(−(1+x)t) ]_0 ^(+∞)   =(1/(2n+1+x))−(1/(1+x)) ⇒f(x) =∫ (dx/(x+2n+1))−∫ (dx/(x+1)) +c  =ln(((x+2n+1)/(x+1))) +c   we have c =lim_(x→+∞) f(x) =0 ⇒  f(x) =ln(((x+2n+1)/(x+1)))  and U_n =f(0) =ln(2n+1)  lim_(n→+∞)  U_n =+∞ ⇒ Σ U_n  is divergent