In[]:=
B[x_,k_]:=Gamma[x+k-1]/(Gamma[x]*Gamma[k])
In[]:=
S[k_,min_,max_,p_]:=B[max/p,k]*p-B[min/p,k]*p
In[]:=
(*Simetría*)​​FullSimplify[B[x,k]==B[k,x]]​​(*Recurrencia*)​​FullSimplify[B[x,k]==(B[x-1,k]+B[x,k-1])]
Out[]=
True
Out[]=
True
In[]:=
(*k=1*)​​FullSimplify[B[x,1]](*1*)​​(*k=2*)​​FullSimplify[B[x,2]](*x*)​​(*k=3*)​​FullSimplify[B[x,3]](*x(x+1)/2*)
Out[]=
1
Out[]=
x
Out[]=
1
2
x(1+x)
In[]:=
Assuming[{a>0,b>0,a<b},Simplify[​​Sum[(a/b)^n,{n,0,Infinity}]​​]]
Out[]=
-
b
a-b
In[]:=
Simplify[Sum[r^n,{n,0,x}]]
Out[]=
-1+
1+x
r
-1+r
In[]:=
DD[f_][x_]:=f[x]-f[x-1]
In[]:=
DI[f_][x_]:=Sum[f[n],{n,1,x}]
In[]:=
Simplify[DI[DD[Function[n,n]]][x]]
Out[]=
x
In[]:=
Simplify[DI[Function[n,n]][x]]
Out[]=
1
2
x(1+x)
In[]:=
Simplify[DD[Function[n,r^n]][x]]
Out[]=
(-1+r)
-1+x
r
In[]:=
Simplify[DI[Function[n,r^n]][x]]
Out[]=
r(-1+
x
r
)
-1+r