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

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume (Partial_Sums s) . n = 2 - ((2 + n) * ((1 / 2) |^ n)) ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = (2 - ((2 + n) * ((1 / 2) |^ n))) + (s . (n + 1)) by SERIES_1:def 1
.= (2 - ((2 + n) * ((1 / 2) |^ n))) + ((n + 1) * ((1 / 2) |^ (n + 1))) by A1
.= (((2 - (2 * ((1 / 2) |^ n))) - (n * ((1 / 2) |^ n))) + (n * ((1 / 2) |^ (n + 1)))) + (1 * ((1 / 2) |^ (n + 1)))
.= (((2 - (((4 * 1) / 2) * ((1 / 2) |^ n))) - (n * ((1 / 2) |^ n))) + (n * (((1 / 2) |^ n) * (1 / 2)))) + (1 * ((1 / 2) |^ (n + 1))) by NEWTON:6
.= (((2 - (4 * (((1 / 2) |^ n) * (1 / 2)))) - (n * ((1 / 2) |^ n))) + ((n * ((1 / 2) |^ n)) * (1 / 2))) + (1 * ((1 / 2) |^ (n + 1)))
.= (((2 - (4 * ((1 / 2) |^ (n + 1)))) - (n * ((1 / 2) |^ n))) + ((n * ((1 / 2) |^ n)) * (1 / 2))) + (1 * ((1 / 2) |^ (n + 1))) by NEWTON:6
.= (2 - (3 * ((1 / 2) |^ (n + 1)))) - (n * (((1 / 2) |^ n) * (1 / 2)))
.= (2 - (3 * ((1 / 2) |^ (n + 1)))) - (n * ((1 / 2) |^ (n + 1))) by NEWTON:6
.= 2 - ((3 + n) * ((1 / 2) |^ (n + 1))) ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= 0 * ((1 / 2) |^ 0) by A1
.= 2 - ((2 + 0) * 1)
.= 2 - ((2 + 0) * ((1 / 2) |^ 0)) by NEWTON:4 ;
then A3: S₁[ 0 ] ;
for n being Nat holds S₁[n] from NAT_1:sch 2(A3, A2);
hence (Partial_Sums s) . n = 2 - ((2 + n) * ((1 / 2) |^ n)) ; :: thesis: verum