let n be Nat; :: thesis: for s being Real_Sequence st ( for n being Nat holds s . n = ((2 * n) - 1) + ((1 / 2) |^ n) ) holds
(Partial_Sums s) . n = ((n |^ 2) + 1) - ((1 / 2) |^ n)
let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n = ((2 * n) - 1) + ((1 / 2) |^ n) ) implies (Partial_Sums s) . n = ((n |^ 2) + 1) - ((1 / 2) |^ n) )
defpred S₁[ Nat] means (Partial_Sums s) . $1 = (($1 |^ 2) + 1) - ((1 / 2) |^ $1);
assume A1: for n being Nat holds s . n = ((2 * n) - 1) + ((1 / 2) |^ n) ; :: thesis: (Partial_Sums s) . n = ((n |^ 2) + 1) - ((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 = ((n |^ 2) + 1) - ((1 / 2) |^ n) ; :: thesis: S₁[n + 1]
then (Partial_Sums s) . (n + 1) = (((n |^ 2) + 1) - ((1 / 2) |^ n)) + (s . (n + 1)) by SERIES_1:def 1
.= (((n |^ 2) + 1) - ((1 / 2) |^ n)) + (((2 * (n + 1)) - 1) + ((1 / 2) |^ (n + 1))) by A1
.= (((((n |^ 2) + 1) - ((1 / 2) |^ n)) + ((1 / 2) |^ (n + 1))) + (2 * (n + 1))) - 1
.= (((((n |^ 2) + 1) - ((1 / 2) |^ n)) + (((1 / 2) |^ n) * (1 / 2))) + (2 * (n + 1))) - 1 by NEWTON:6
.= ((((n |^ 2) + 1) - (((1 / 2) |^ n) * (1 / 2))) + (2 * (n + 1))) - 1
.= ((((n |^ 2) + 1) - ((1 / 2) |^ (n + 1))) + (2 * (n + 1))) - 1 by NEWTON:6
.= ((((n |^ 2) + (2 * n)) + 1) + 1) - ((1 / 2) |^ (n + 1))
.= ((((n |^ 2) + ((2 * n) * 1)) + (1 |^ 2)) + 1) - ((1 / 2) |^ (n + 1))
.= (((n + 1) |^ 2) + 1) - ((1 / 2) |^ (n + 1)) by Lm1 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= ((2 * 0) - 1) + ((1 / 2) |^ 0) by A1
.= (- 1) + 1 by NEWTON:4
.= (1 - 1) + (0 |^ 2) by NEWTON:11
.= (1 - ((1 / 2) |^ 0)) + (0 |^ 2) 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 = ((n |^ 2) + 1) - ((1 / 2) |^ n) ; :: thesis: verum