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1.1!+2.2!+3.3!+4.4!+5.5!
=(2-1).1!+(3-1).2!+(4-1).3!+
(5-1).4!+(6-1).5!
=2!-1!+3!-2!+4!-3!+5!-4!+6!-5!
=6!-1!=720-1=719
\(1.1!+2.2!+3.3!+4.4!+5.5!\)
\(=\left(2-1\right)1!+\left(3-1\right)2!+....+\left(6-1\right)5!\)
\(=2!-1!+3!-2!+4!-3!+5!-4!+6!-5!\)
\(=6!-1\)
\(=720-1\)
\(=719\)

S = 1.1! + 2.2! + 3.3! + 4.4! + 5.5! = 1 + 2.1.2 + 3.1.2.3 + 4.1.2.3.4 + 5.1.2.3.4.5 = 1 + 4 + 18 + 96 + 600 = 719

Ta có : \(n.n!=\left(n+1-1\right).n!=n!.\left(n+1\right)-n!=\left(n+1\right)!-n!\)
Áp dụng : \(1.1!+2.2!+3.3!+4.4!+5.5!=2!-1!+3!-2!+4!-3!+5!-4!+6!-5!\)
\(=6!-1=1.2.3.4.5.6-1=719\)


ta có công thức quy lạp \(1.1!+2.2!+...+n.n!=\left(n+1\right)!-1\)
áp dụng vào bài \(1.1!+2.2!+3.3!+...+7.7!=\left(7+1\right)!-1=8!-1=40320-1=40319\)

1.1!+2.2!+3.3!+4.4!+5.5!
=(2-1).1!+(3-1).2!+(4-1).3!+
(5-1).4!+(6-1).5!
=2!-1!+3!-2!+4!-3!+5!-4!+6!-5!
=6!-1!=720-1=719

chung to :C = \(\frac{1}{1.1!}+\frac{1}{2.2!}+\frac{1}{3.3!}+...+\frac{1}{2019.2019!}< \frac{3}{2}\)

Thấy : \(\frac{1}{1.1!}=\frac{1}{1}\)
\(\frac{1}{2.2!}=\frac{1}{4}\)
\(\frac{1}{3.3!}< \frac{1}{1.2.3}\)( Vì 3.3! > 1.2.3 )
...
\(\frac{1}{2019.2019!}< \frac{1}{2017.2018.2019}\)( vì 2019.2019! < 2017.2018.2019)
Cộng từng vế có :
\(\frac{1}{3.3!}+\frac{1}{4.4!}+...+\frac{1}{2019.2019!}< \frac{1}{1.2.3}+...+\frac{1}{2017.2018.2019}\)
\(\Rightarrow\frac{1}{1.1!}+\frac{1}{2.2!}+...+\frac{1}{2019.2019!}< \frac{1}{1}+\frac{1}{4}+\frac{1}{1.2.3}+...+\frac{1}{2017.2018.2019}\)
\(\Rightarrow C< \frac{1}{1}+\frac{1}{4}+\left(\frac{1}{1.2}-\frac{1}{2.3}+...+\frac{1}{2017.2018}-\frac{1}{2018.2019}\right):2\)
\(\Rightarrow C< \frac{1}{1}+\frac{1}{4}+\left(\frac{1}{2}-\frac{1}{2018.2019}\right):2\)
\(\Rightarrow C< \frac{3}{2}-\frac{1}{2.2018.2019}\)
Vì \(\frac{1}{2.2018.2019}>0\Rightarrow C< \frac{3}{2}\)
Ta có: \(1\cdot1+2\cdot2+\cdots+2021\cdot2021\)
\(=1^2+2^2+\cdots+2021^2\)
\(=2021\cdot\frac{\left(2021+1\right)\left(2\cdot2021+1\right)}{6}=2021\cdot2022\cdot\frac{4043}{6}\)
=2753594311