With[{n = 8}, 
      FindRoot[0.1 == 
          v (∑_(i = 0)^n)((((1 - (((v))^2)))^i)/((2 i + 1) Beta[1 + i, (1/2)])), {v, 0.9}]]

InverseCDFModNorm[s_, y_] := With[{x = y*Beta[1 + s, 1/2]}, x + (s*x^3)/3 + (1/30)*(3*s + 7*s^2)*x^5 + (1/630)*(30*s + 123*s^2 + 127*s^3)*x^7 + 
    ((630*s + 3579*s^2 + 6822*s^3 + 4369*s^4)*x^9)/22680 + ((22680*s + 162630*s^2 + 439899*s^3 + 532542*s^4 + 243649*s^5)*x^11)/1247400 + 
    ((1247400*s + 10735470*s^2 + 37155267*s^3 + 64697499*s^4 + 56717781*s^5 + 20036983*s^6)*x^13)/97297200 + 
    ((97297200*s + 973580580*s^2 + 4079027880*s^3 + 9165745647*s^4 + 11656842609*s^5 + 7959325221*s^6 + 2280356863*s^7)*x^15)/10216206000 + 
    ((10216206000*s + 116270210700*s^2 + 569674791180*s^3 + 1558544277681*s^4 + 2572662311496*s^5 + 2563294235106*s^6 + 1427877062076*s^7 + 343141433761*s^8)*x^17)/
     1389404016000]