∀n ≥ 1, Xn i=1 i(i!) = (n + 1)! − 1
∑i=1ni(i!)=(n+1)!−1\displaystyle{\sum^n_{i=1} }i(i!)=(n+1)!-1i=1∑ni(i!)=(n+1)!−1
proof by induction:
for n=1:
∑i=111⋅(1!)=1\displaystyle{\sum^1_{i=1} }1\cdot(1!)=1i=1∑11⋅(1!)=1
(1+1)!−1=1(1+1)!-1=1(1+1)!−1=1
let for n=k:
∑i=1ki(i!)=(k+1)!−1\displaystyle{\sum^k_{i=1} }i(i!)=(k+1)!-1i=1∑ki(i!)=(k+1)!−1
then for n=k+1:
∑i=1k+1i(i!)=∑i=1ki(i!)+(k+1)(k+1)!=(k+1)!−1+(k+1)(k+1)!=\displaystyle{\sum^{k+1}_{i=1} }i(i!)=\displaystyle{\sum^{k}_{i=1} }i(i!)+(k+1)(k+1)!=(k+1)!-1+(k+1)(k+1)!=i=1∑k+1i(i!)=i=1∑ki(i!)+(k+1)(k+1)!=(k+1)!−1+(k+1)(k+1)!=
=(k+2)(k+1)!−1=(k+2)!−1=(k+2)(k+1)!-1=(k+2)!-1=(k+2)(k+1)!−1=(k+2)!−1
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