let a be Real; :: thesis: for s being Real_Sequence st a <> 1 & ( for n being Nat holds s . n = n * (a |^ n) ) holds
for n being Nat holds () . n = ((a * (1 - (a |^ n))) / ((1 - a) |^ 2)) - ((n * (a |^ (n + 1))) / (1 - a))

let s be Real_Sequence; :: thesis: ( a <> 1 & ( for n being Nat holds s . n = n * (a |^ n) ) implies for n being Nat holds () . n = ((a * (1 - (a |^ n))) / ((1 - a) |^ 2)) - ((n * (a |^ (n + 1))) / (1 - a)) )
assume that
A1: a <> 1 and
A2: for n being Nat holds s . n = n * (a |^ n) ; :: thesis: for n being Nat holds () . n = ((a * (1 - (a |^ n))) / ((1 - a) |^ 2)) - ((n * (a |^ (n + 1))) / (1 - a))
defpred S1[ Nat] means () . \$1 = ((a * (1 - (a |^ \$1))) / ((1 - a) |^ 2)) - ((\$1 * (a |^ (\$1 + 1))) / (1 - a));
A3: 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 = ((a * (1 - (a |^ n))) / ((1 - a) |^ 2)) - ((n * (a |^ (n + 1))) / (1 - a)) ; :: thesis: S1[n + 1]
then () . (n + 1) = (((a * (1 - (a |^ n))) / ((1 - a) |^ 2)) - ((n * (a |^ (n + 1))) / (1 - a))) + (s . (n + 1)) by SERIES_1:def 1
.= (((a * (1 - (a |^ n))) / ((1 - a) |^ 2)) - ((n * (a |^ (n + 1))) / (1 - a))) + ((n + 1) * (a |^ (n + 1))) by A2
.= ((((a * (1 - (a |^ n))) / ((1 - a) |^ 2)) - ((n * (a |^ (n + 1))) / (1 - a))) + (n * (a |^ (n + 1)))) + (a |^ (n + 1))
.= ((a * (1 - (a |^ (n + 1)))) / ((1 - a) |^ 2)) - (((n + 1) * (a |^ (n + 2))) / (1 - a)) by ;
hence S1[n + 1] ; :: thesis: verum
end;
() . 0 = s . 0 by SERIES_1:def 1
.= 0 * (a |^ 0) by A2
.= (((1 - 1) * a) / ((1 - a) |^ 2)) - ((0 * (a |^ (0 + 1))) / (1 - a))
.= (((1 - (a |^ 0)) * a) / ((1 - a) |^ 2)) - ((0 * (a |^ (0 + 1))) / (1 - a)) by NEWTON:4 ;
then A4: S1[ 0 ] ;
for n being Nat holds S1[n] from NAT_1:sch 2(A4, A3);
hence for n being Nat holds () . n = ((a * (1 - (a |^ n))) / ((1 - a) |^ 2)) - ((n * (a |^ (n + 1))) / (1 - a)) ; :: thesis: verum