let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = ((n + 1) * (2 |^ n)) / ((n + 2) * (n + 3)) ) implies for n being Nat holds (Partial_Sums s) . n = ((2 |^ (n + 1)) / (n + 3)) - (1 / 2) )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = ((2 |^ ($1 + 1)) / ($1 + 3)) - (1 / 2);
assume A1: for n being Nat holds s . n = ((n + 1) * (2 |^ n)) / ((n + 2) * (n + 3)) ; :: thesis: for n being Nat holds (Partial_Sums s) . n = ((2 |^ (n + 1)) / (n + 3)) - (1 / 2)
A2: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
n + 3 >= 3 by NAT_1:11;
then A3: n + 3 > 0 by XXREAL_0:2;
n + 4 >= 4 by NAT_1:11;
then A4: n + 4 > 0 by XXREAL_0:2;
assume (Partial_Sums s) . n = ((2 |^ (n + 1)) / (n + 3)) - (1 / 2) ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = (((2 |^ (n + 1)) / (n + 3)) - (1 / 2)) + (s . (n + 1)) by SERIES_1:def 1
.= (((2 |^ (n + 1)) / (n + 3)) - (1 / 2)) + ((((n + 1) + 1) * (2 |^ (n + 1))) / (((n + 1) + 2) * ((n + 1) + 3))) by A1
.= (((2 |^ (n + 1)) / (n + 3)) + (((n + 2) * (2 |^ (n + 1))) / ((n + 3) * (n + 4)))) - (1 / 2)
.= ((((2 |^ (n + 1)) * (n + 4)) / ((n + 3) * (n + 4))) + (((n + 2) * (2 |^ (n + 1))) / ((n + 3) * (n + 4)))) - (1 / 2) by A4, XCMPLX_1:91
.= ((((2 |^ (n + 1)) * (n + 4)) + ((n + 2) * (2 |^ (n + 1)))) / ((n + 3) * (n + 4))) - (1 / 2) by XCMPLX_1:62
.= ((((2 |^ (n + 1)) * 2) * (n + 3)) / ((n + 4) * (n + 3))) - (1 / 2)
.= (((2 |^ (n + 1)) * 2) / (n + 4)) - (1 / 2) by A3, XCMPLX_1:91
.= ((2 |^ ((n + 1) + 1)) / ((n + 1) + 3)) - (1 / 2) by NEWTON:6 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= ((0 + 1) * (2 |^ 0)) / ((0 + 2) * (0 + 3)) by A1
.= (1 * 1) / 6 by NEWTON:4
.= (2 / 3) - (1 / 2)
.= ((2 |^ (0 + 1)) / (0 + 3)) - (1 / 2) ;
then A5: S₁[ 0 ] ;
for n being Nat holds S₁[n] from NAT_1:sch 2(A5, A2);
hence for n being Nat holds (Partial_Sums s) . n = ((2 |^ (n + 1)) / (n + 3)) - (1 / 2) ; :: thesis: verum