Chứng minh rằng 1.2-1/2! + 2.3-1/3! + 3.4-1?4! +...+ 99.100-1/100! <2
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MM
1
23 tháng 6 2016
\(\frac{1.2-1}{2!}+\frac{2.3-1}{3!}+............+\frac{99.100-1}{100!}\)
\(=\frac{1.2}{2!}-\frac{1}{2!}+\frac{2.3}{3!}-\frac{1}{3!}+..........+\frac{99.100}{100!}-\frac{1}{100!}\)
\(=\left(\frac{1.2}{2!}+\frac{2.3}{3!}+.........+\frac{99.100}{100!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+.....+\frac{1}{100!}\right)\)
\(=\left(1+1+\frac{1}{2!}+.........+\frac{1}{98!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+....+\frac{1}{100!}\right)\)
\(=2-\frac{1}{99!}-\frac{1}{100!}< 2\)
\(\Rightarrowđpcm\)
VM
6 tháng 1 2020
Đặt \(A=\frac{1.2-1}{2!}+\frac{2.3-1}{3!}+\frac{3.4-1}{4!}+...+\frac{99.100-1}{100!}\)
\(\Rightarrow A=\frac{1.2}{2!}-\frac{1}{2!}+\frac{2.3}{3!}-\frac{1}{3!}+...+\frac{99.100}{100!}-\frac{1}{100!}\)
\(\Rightarrow A=\left(\frac{1.2}{2!}+\frac{2.3}{3!}+\frac{3.4}{4!}+...+\frac{99.100}{100!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{100!}\right)\)
\(\Rightarrow A=\left(1+1+\frac{1}{2!}+...+\frac{1}{98!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+\frac{1}{\text{4!}}+...+\frac{1}{100!}\right)\)
\(\Rightarrow A=1+1-\frac{1}{99!}-\frac{1}{100!}\)
\(\Rightarrow A=2-\frac{1}{99!}-\frac{1}{100!}\)
Mà \(2-\frac{1}{99!}-\frac{1}{100!}< 2.\)
\(\Rightarrow A< 2\left(đpcm\right).\)
Chúc bạn học tốt!
24 tháng 6 2019
A= \(\frac{1}{2}\) + \(\frac{1}{2^2}\) + \(\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{2^{100}}\)
\(\Rightarrow\) 2A = 1 + \(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{2^{99}}\)
\(\Rightarrow\) 2A - A = ( \(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{2^{100}}\) ) -
( \(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+...+\frac{1}{2^{99}}\))
\(\Rightarrow\) A = 1 - \(\frac{1}{2^{100}}\) < 1
Vậy: A < 1
\(\frac{1}{2}\)
24 tháng 6 2019
B= \(\frac{2}{1.2}+\frac{2}{2.3}+\frac{2}{3.4}+...+\frac{2}{99.100}\)
= 2. \(\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\right)\)
= 2. ( \(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\) )
= 2. \(\left(\frac{1}{1}-\frac{1}{100}\right)\) = \(\frac{99}{50}\)
\(\Rightarrow\) B = \(\frac{99}{50}\) < \(\frac{100}{50}\) = 2
Vậy: B < 2
KM
6
CH
Cô Hoàng Huyền
Admin
VIP
21 tháng 6 2016
Ta thấy mỗi hạng tử của tổng đều có dạng: \(\frac{\left(n-1\right)n-1}{n!}=\frac{\left(n-1\right)n}{n!}-\frac{1}{n!}=\frac{1}{\left(n-2\right)!}-\frac{1}{n!}\)
Như vậy VT = \(\frac{1}{0!}-\frac{1}{2!}+\frac{1}{1!}-\frac{1}{3!}+\frac{1}{2!}-\frac{1}{4!}+\frac{1}{3!}-\frac{1}{5!}+...+\frac{1}{98!}-\frac{1}{100!}\)
\(=2-\frac{1}{99!}-\frac{1}{100!}< 2\)
9 tháng 6 2017
sửa đề câu 1 :
\(\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+...+\frac{99}{100!}\)
\(=\frac{2-1}{2!}+\frac{3-1}{3!}+\frac{4-1}{4!}+...+\frac{100-1}{100!}\)
\(=\frac{1}{1!}-\frac{1}{2!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{3!}-\frac{1}{4!}+...+\frac{1}{99!}-\frac{1}{100!}\)
\(=1-\frac{1}{100!}< 1\)
sửa đề câu 2
\(\frac{1.2-1}{2!}+\frac{2.3-1}{3!}+\frac{3.4-1}{4!}+...+\frac{99.100-1}{100!}\)
\(=\frac{1.2}{2!}-\frac{1}{2!}+\frac{2.3}{3!}-\frac{1}{3!}+\frac{3.4}{4!}-\frac{1}{4!}+...+\frac{99.100}{100!}-\frac{1}{100!}\)
\(=\left(\frac{1.2}{2!}+\frac{2.3}{3!}+\frac{3.4}{4!}+...+\frac{99.100}{100!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+...+\frac{1}{100!}\right)\)
\(=\left(1+1+\frac{1}{2!}+...+\frac{1}{98!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+...+\frac{1}{100!}\right)\)
\(=2-\frac{1}{99!}-\frac{1}{100!}< 2\)
23 tháng 10 2016
Chứng minh rằng:
\(\frac{1.2-1}{2!}+\frac{2.3-1}{3!}+\frac{3.4-1}{4!}+...+\frac{99.100-1}{100!}< 2\)
Ta có:
\(\frac{1.2-1}{2!}+\frac{2.3-1}{3!}+\frac{3.4-1}{4!}+...+\frac{99.100-1}{100!}\\ =\frac{1.2}{2!}-\frac{1}{2!}+\frac{2.3}{3!}-\frac{1}{3!}+\frac{3.4}{4!}-\frac{1}{4!}+...+\frac{99.100}{100!}-\frac{1}{100!}\)
\(=\left(\frac{1.2}{2!}+\frac{2.3}{3!}+\frac{3.4}{4!}+...+\frac{99.100}{100!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+...+\frac{1}{100!}\right)\)
\(=\left(1+1+\frac{1}{2!}+...+\frac{1}{98!}\right)-\left(\frac{1}{2!}+\frac{1}{3!}+...+\frac{1}{100!}\right)\)
\(=2-\frac{1}{99!}-\frac{1}{100}< 2\)
13 tháng 11 2017
1.
\(\dfrac{1}{2!}+\dfrac{2}{3!}+\dfrac{3}{4!}+...+\dfrac{99}{100!}\)
\(=\dfrac{2-1}{2!}+\dfrac{3-1}{3!}+\dfrac{4-1}{4!}+...+\dfrac{100-1}{100!}\)
\(=\dfrac{1}{1!}-\dfrac{1}{2!}+\dfrac{1}{2!}-\dfrac{1}{3!}+\dfrac{1}{3!}-\dfrac{1}{4!}+...+\)\(\dfrac{1}{99!}-\dfrac{1}{100!}\)
\(=1-\dfrac{1}{100!}< 1\)
13 tháng 11 2017
2.
\(\dfrac{1.2-1}{2!}+\dfrac{2.3-1}{3!}+\dfrac{3.4-1}{4!}+...+\)\(\dfrac{1}{100!}\)
Ta có:
\(=\dfrac{1.2}{2!}-\dfrac{1}{2!}+\dfrac{2.3}{3!}-\dfrac{1}{3!}+\dfrac{3.4}{4!}-\dfrac{1}{4!}+...+\)\(\dfrac{99.100}{100!}-\dfrac{1}{100}\)
\(=\left(\dfrac{1.2}{2!}+\dfrac{2.3}{3!}+\dfrac{3.4}{4!}+...+\dfrac{99.100}{100!}\right)\)\(-\left(\dfrac{1}{2!}+\dfrac{1}{3!}+...+\dfrac{1}{100!}\right)\)
\(=\left(1+1+\dfrac{1}{2!}+...+\dfrac{1}{98!}\right)\)\(-\left(\dfrac{1}{2!}+\dfrac{1}{3!}+...+\dfrac{1}{100!}\right)\)
\(=2-\dfrac{1}{99!}-\dfrac{1}{100!}< 2\)
3 tháng 4 2017
Ta có:
\(\dfrac{1.2-1}{2!}+\dfrac{2.3-1}{3!}+...+\dfrac{99.100-1}{100!}\)
\(=\dfrac{1.2}{2!}-\dfrac{1}{2!}+\dfrac{2.3}{3!}-\dfrac{1}{3!}+...+\dfrac{99.100}{100!}-\dfrac{1}{100!}\)
\(=\left(\dfrac{1.2}{2!}+\dfrac{2.3}{3!}+...+\dfrac{99.100}{100!}\right)-\left(\dfrac{1}{2!}+\dfrac{1}{3!}+...+\dfrac{1}{100!}\right)\)
\(=\left(1+1+\dfrac{1}{2!}+...+\dfrac{1}{98!}\right)-\left(\dfrac{1}{2!}+\dfrac{1}{3!}+...+\dfrac{1}{100!}\right)\)
\(=1+1-\dfrac{1}{99!}-\dfrac{1}{100!}\)
\(=2-\dfrac{1}{99!}-\dfrac{1}{100!}< 2\)
Vậy \(\dfrac{1.2-1}{2!}+\dfrac{2.3-1}{3!}+...+\dfrac{99.100-1}{100!}< 2\) (Đpcm)