let a be real number ; :: thesis: for s being Real_Sequence st a <> 1 & ( for n being Element of NAT holds s . n = n * (a |^ n) ) holds
for n being Element of NAT holds (Partial_Sums s) . 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 Element of NAT holds s . n = n * (a |^ n) ) implies for n being Element of NAT holds (Partial_Sums s) . n = ((a * (1 - (a |^ n))) / ((1 - a) |^ 2)) - ((n * (a |^ (n + 1))) / (1 - a)) )
assume that
A1: a <> 1 and
A2: for n being Element of NAT holds s . n = n * (a |^ n) ; :: thesis: for n being Element of NAT holds (Partial_Sums s) . n = ((a * (1 - (a |^ n))) / ((1 - a) |^ 2)) - ((n * (a |^ (n + 1))) / (1 - a))
defpred S₁[ Element of NAT ] means (Partial_Sums s) . $1 = ((a * (1 - (a |^ $1))) / ((1 - a) |^ 2)) - (($1 * (a |^ ($1 + 1))) / (1 - a));
A3: for n being Element of NAT st S₁[n] holds
S₁[n + 1]

proof

let n be Element of 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 (Partial_Sums s) . (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 A1, Lm6 ;
hence S₁[n + 1] ; :: thesis: verum

end;

(Partial_Sums s) . 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 Element of NAT holds S₁[n] from NAT_1:sch 1(A4, A3);
hence for n being Element of NAT holds (Partial_Sums s) . n = ((a * (1 - (a |^ n))) / ((1 - a) |^ 2)) - ((n * (a |^ (n + 1))) / (1 - a)) ; :: thesis: verum