//forn>=0,Binomial[-n,k]=
k
(-1)
Binomial[n+k-1,k]
//so,Binomial[-3,k]=
k
(-1)
Binomial[2+k,k]=
k
(-1)
(1/2)(2+k)(1+k)
In[]:=
tt[k_,s_]:=8
k+1
(-1)
Binomial[-3,k]Sin[2s(3/2+k)]
In[]:=
tt2[k_,s_]:=-4(2+k)(1+k)Sin[2s(3/2+k)]
In[]:=
N[tt[4,3]]
Out[]=
-119.989
In[]:=
N[tt2[4,3]]
Out[]=
-119.989
In[]:=
N[tt[4000,3]]
Out[]=
-5.20582×
7
10
In[]:=
N[tt2[4000,3]]
Out[]=
-5.20582×
7
10
In[]:=
si2[s_]:=8Sum[
k+1
(-1)
Binomial[-3,k]Sin[2s(3/2+k)],{k,0,Infinity}]
In[]:=
N[si2[3]]
Out[]=
355.823
In[]:=
si1[s_]:=
-3
Sin[s]
In[]:=
si[s_]:=4Sum[-(2+k)(1+k)Sin[2s(3/2+k)],{k,0,Infinity}]
In[]:=
N[si[3]]
Sum
:Sum does not converge.
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in k near {k} = {4.81037×
7
10
}. NIntegrate obtained -1.40845×
10
10
and 1.40311×
10
10
for the integral and error estimates.
Out[]=
-5.6338×
10
10