Ta có : 

\(S=\frac{2}{3!}+\frac{3}{4!}+\frac{4}{5!}+...+\frac{2016}{2017!}\)

\(S=\frac{3-1}{3!}+\frac{4-1}{4!}+\frac{5-1}{5!}+...+\frac{2017-1}{2017!}\)

\(S=\frac{3}{3!}-\frac{1}{3!}+\frac{4}{4!}-\frac{1}{4!}+\frac{5}{5!}-\frac{1}{5!}+...+\frac{2017}{2017!}-\frac{1}{2017!}\)

\(S=\frac{1}{2!}-\frac{1}{3!}+\frac{1}{3!}-\frac{1}{4!}+\frac{1}{4!}-\frac{1}{5!}+...+\frac{1}{2016!}-\frac{1}{2017!}\)

\(S=\frac{1}{2!}-\frac{1}{2017!}\)

\(S=\frac{1}{2}-\frac{1}{2017!}\)

Vậy \(S=\frac{1}{2}-\frac{1}{2017!}\)

Chúc bạn học tốt ~ 

So sánh với \(\frac{1}{2}\)

Mấy bài dạng này biết cách làm là oke 

Ta có : 

\(A=\frac{\frac{2016}{1}+\frac{2015}{2}+\frac{2014}{3}+...+\frac{2}{2015}+\frac{1}{2016}}{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2016}+\frac{1}{2017}}\)

\(A=\frac{\left(2016-1-1-...-1\right)+\left(\frac{2015}{2}+1\right)+\left(\frac{2014}{3}+1\right)+...+\left(\frac{2}{2015}+1\right)+\left(\frac{1}{2016}+1\right)}{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2016}+\frac{1}{2017}}\)

\(A=\frac{\frac{2017}{2017}+\frac{2017}{2}+\frac{2017}{3}+...+\frac{2017}{2015}+\frac{2017}{2016}}{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2016}+\frac{1}{2017}}\)

\(A=\frac{2017\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2016}+\frac{1}{2017}\right)}{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2016}+\frac{1}{2017}}\)

\(A=2017\)

Vậy \(A=2017\)

Chúc bạn học tốt ~ 

\(A=\frac{\frac{2016}{1}+\frac{2015}{2}+...+\frac{2}{2015}+\frac{1}{2016}}{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2017}}\)

\(A=\frac{2016+\frac{2015}{2}+...+\frac{2}{2015}+\frac{1}{2016}}{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2017}}\)

\(A=\frac{\left(\frac{2015}{2}+1\right)+\left(\frac{2014}{3}+1\right)+...+\left(\frac{2}{2015}+1\right)+\left(\frac{1}{2016}+1\right)+\frac{2017}{2017}}{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2017}}\)

(số 2016 tách ra làm 2016 số 1 rồi cộng vào từng phân số, còn dư 1 số viết thành 2017/2017 nghe bạn!!! :)))

\(A=\frac{\frac{2017}{2}+\frac{2017}{3}+...+\frac{2017}{2015}+\frac{2017}{2016}+\frac{2017}{2017}}{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2017}}\)

\(A=\frac{2017\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2015}+\frac{1}{2016}+\frac{1}{2017}\right)}{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2017}}\)

\(A=2017\)

 P \(=\left(1-\frac{1}{2^2}\right).\left(1-\frac{1}{3^2}\right).\left(1-\frac{1}{4^2}\right)...\left(1-\frac{1}{50^2}\right)\) 

P\(=\frac{2^2-1}{2^2}.\frac{3^2-1}{3^2}.\frac{4^2-1}{4^2}...\frac{50^2-1}{50^2}\)

P \(=\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}...\frac{49.51}{50.50}\)

P\(=\frac{\left(1.2.3...49\right).\left(3.4.5...51\right)}{\left(2.3.4...50\right).\left(2.3.4...50\right)}\)

P\(=\frac{1.51}{50.2}=\frac{51}{100}\)

\(A< \frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{2015.2016}=1-\frac{1}{2016}=\frac{2015}{2016}< \frac{1512}{2016}=\frac{3}{4}\)

Vậy \(A< \frac{3}{4}\)

bài này dễ mà

Ta có:

S = 1/2² + 1/3² + 1/4² + ... + 1/2016²

    = 1/2.2 + 1/3.3 + 1/4.4 + ... + 1/2016.2016

S  < 1/1.2 + 1/2.3 + 1/3.4 + ... + 1/2015.2016

S  < 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/2015 - 1/2016

S  < 1 - 1/2016

Mà 1 - 1/2016 < 1

=> S < 1

Vậy S < 1

Ủng hộ nha