In[]:=
Sum[Sum[((-1)^kLegendreP[n,k,2,-3/5])/((1-(-3/5)^2)^(k/2)k!)(n+1/2)LegendreP[n,3/5]Integrate[t(6/5(t+1)^(t+1)/(t+2))^kD[6/5(t+1)^(t+1)/(t+2)-3/5,t],{t,0,1}],{k,0,n}],{n,0,3}]
Out[]=
-
306
625
1
∫
0
t-
6
1+t
(1+t)
5
2
(2+t)
+
6
1+t
(1+t)
(1+Log[1+t])
5(2+t)
t-
63
20
1
∫
0
216t
3+3t
(1+t)
-
6
1+t
(1+t)
5
2
(2+t)
+
6
1+t
(1+t)
(1+Log[1+t])
5(2+t)
125
3
(2+t)
t+
291
50
1
∫
0
36t
2+2t
(1+t)
-
6
1+t
(1+t)
5
2
(2+t)
+
6
1+t
(1+t)
(1+Log[1+t])
5(2+t)
25
2
(2+t)
t-
99
125
1
∫
0
6t
1+t
(1+t)
-
6
1+t
(1+t)
5
2
(2+t)
+
6
1+t
(1+t)
(1+Log[1+t])
5(2+t)
5(2+t)
t
In[]:=
Sum[Sum[((-1)^kLegendreP[n,k,2,-3/5])/((1-(-3/5)^2)^(k/2)k!)(n+1/2)LegendreP[n,3/5]NIntegrate[t(6/5(t+1)^(t+1)/(t+2))^kD[6/5(t+1)^(t+1)/(t+2)-3/5,t],{t,0,1}],{k,0,n}],{n,0,28}]
Out[]=
-0.648316
In[]:=
Sum[Sum[((-1)^kLegendreP[n,k,2,-3/5])/((1-(-3/5)^2)^(k/2)k!)(n+1/2)LegendreP[n,3/5]NIntegrate[t(6/5(t+1)^(t+1)/(t+2))^kD[6/5(t+1)^(t+1)/(t+2)-3/5,t],{t,0,1}],{k,0,27}],{n,0,27}]
Out[]=
1.26721
In[]:=
Sum[Sum[((-1)^kLegendreP[n,k,2,-3/5])/((1-(-3/5)^2)^(k/2)k!)(n+1/2)LegendreP[n,3/5]NIntegrate[t(6/5(t+1)^(t+1)/(t+2))^kD[6/5(t+1)^(t+1)/(t+2)-3/5,t],{t,0,1}],{k,0,26}],{n,0,26}]
Out[]=
0.780639
In[]:=
Sum[Sum[((-1)^kLegendreP[n,k,2,-3/5])/((1-(-3/5)^2)^(k/2)k!)(n+1/2)LegendreP[n,3/5]NIntegrate[t(6/5(t+1)^(t+1)/(t+2))^kD[6/5(t+1)^(t+1)/(t+2)-3/5,t],{t,0,1}],{k,0,n}],{n,0,25}]
Out[]=
0.780791
In[]:=
Sum[Sum[((-1)^kLegendreP[n,k,2,-3/5])/((1-(-3/5)^2)^(k/2)k!)(n+1/2)LegendreP[n,3/5]NIntegrate[t(6/5(t+1)^(t+1)/(t+2))^kD[6/5(t+1)^(t+1)/(t+2)-3/5,t],{t,0,1}],{k,0,n}],{n,0,24}]
Out[]=
0.771697
In[]:=
Plot[Sum[Sum[((-1)^kLegendreP[n,k,2,-3/5])/((1-(-3/5)^2)^(k/2)k!)(n+1/2)LegendreP[n,x]NIntegrate[t(6/5(t+1)^(t+1)/(t+2))^kD[6/5(t+1)^(t+1)/(t+2)-3/5,t],{t,0,1}],{k,0,n}],{n,0,7}],{x,0,1}]
Out[]=
In[]:=
Plot[Sum[Sum[((-1)^kLegendreP[n,k,2,-3/5])/((1-(-3/5)^2)^(k/2)k!)(n+1/2)LegendreP[n,x]NIntegrate[t(6/5(t+1)^(t+1)/(t+2))^kD[6/5(t+1)^(t+1)/(t+2)-3/5,t],{t,0,1}],{k,0,n}],{n,0,10}],{x,0,1}]
Cloud
::timelimit
:This computation has exceeded the time limit for your plan.
​
Sum[(n+1/2)LegendreP[n,3/5]NIntegrate[tLegendreP[n,(6/5(t+1)^(t+1)/(t+2)-3/5)]D[6/5(t+1)^(t+1)/(t+2),t],{t,0,1}],{n,0,25}]
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in t near {t} = {0.929672}. NIntegrate obtained 0.000539183 and 1.01235×
-9
10
for the integral and error estimates.
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in t near {t} = {0.917953}. NIntegrate obtained 0.00144801 and 4.15576×
-9
10
for the integral and error estimates.
NIntegrate
:Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in t near {t} = {0.912094}. NIntegrate obtained -0.00039567 and 1.29325×
-8
10
for the integral and error estimates.
NIntegrate
:Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.
Out[]=
0.775383
In[]:=
Sum[Sum[((-1)^kLegendreP[n,k,2,-3/5])/((1-(-3/5)^2)^(k/2)k!)(n+1/2)LegendreP[n,3/5](6/5)^(k+1)((4/3)^(k+1)/(k+1)-1/(k+1)NIntegrate[(t^t/(t+1))^(k+1),{t,1,2}]),{k,0,n}],{n,0,25}]
Out[]=
0.784583
In[]:=
Sum[Sum[Sum[((-1)^kLegendreP[n,k,2,-3/5])/((1-(-3/5)^2)^(k/2)k!)(n+1/2)LegendreP[n,3/5](6/5)^(k+1)((4/3)^(k+1)/(k+1)-(k+1)^(m-1)/m!NIntegrate[(tLog[t])^m/(t+1)^(k+1),{t,1,2}]),{m,0,20}],{k,0,20}],{n,0,20}]
Out[]=
-574.511
​
In[]:=
ClearAll[f,Fn,xn,x,t,u,DP];DP=9;​​f[x_]:=x^x-x-1;​​Fn[n_]:=Normal@Simplify[InverseSeries@Series[f[x],​​{x,2,n}]]/.{Log[u_]t*Round@N[Log[2,u],DP]};​​xn[n_]:=(xn[n]=N[Fn[n]/.{x0,tLog[2]},DP]);​​x0=x/.FindRoot[f[x],{x,2},WorkingPrecisionDP][[1]];​​Table[{n,xn[n],x0-xn[n]/xn[n-1]},{n,2,9}]//Column
Out[]=
{2,1.79176261,0.79593724}
{3,1.78190466,0.78227686}
{4,1.77867202,0.77858919}
{5,1.77751331,0.77742649}
{6,1.77707297,0.77702277}
{7,1.77689857,0.77687318}
{8,1.77682734,0.77681512}
{9,1.77679756,0.77679180}