Ta có: \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2010^2}

\(A=\frac{2010}{2011}+\frac{2011}{2012}+\frac{2012}{2010}\)

\(A=\frac{4064340600}{4066362660}+\frac{4064341605}{4066362660}+\frac{4070408792}{4066362660}\)

\(A=3,000000742\)

\(B=\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+....+\frac{1}{17}\)

\(B=1,939552553\)

vì đây là so sánh hai dòng phân số nên ta  đổi ra thập phân nhé

do 3,000000742 > 1,939552553 và 3 > 1 Nên A > B nhé

đúng thì k nhé

chúc học giỏi !!!!

 A > B nha bạn !!! ( $ _ $ )%%%

S=\(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{2010.2011.2012}\)

  =\(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{2010.2011}-\frac{1}{2011.2012}\)

  =\(\frac{1}{2}-\frac{1}{2011.2012}< \frac{1}{2}\)(Vì \(\frac{1}{2011.2012}>0\))

=> S <\(\frac{1}{2}\)

\(S=\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+....+\frac{2}{2010.2011.2012}\)

\(S=\frac{3-1}{1.2.3}+\frac{4-2}{2.3.4}+\frac{5-3}{3.4.5}+...+\frac{2012-2010}{2010.2011.2012}\)

\(S=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{2010.2011}-\frac{1}{2011.2012}\)

\(S=\frac{1}{1.2}-\frac{1}{2011.2012}=\frac{2023065}{4046132}\)

\(\text{Vì}\)\(\frac{2023065}{4046132}< \frac{1}{2}\Rightarrow S< P\)

2010+2011=48.50

    dap so 49.50

2010+2011=48.50

k cho mik nhe

là sao mình không hiểu cậu trình bày chi tiết vào chứ mình không được thông minh cho lắm!

Ta có:

\(A=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{2012}}\)

\(\Rightarrow3A=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{2011}}\)

\(\Rightarrow3A-A=\left(1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{2011}}\right)-\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{2012}}\right)\)

\(\Rightarrow2A=1-\frac{1}{3^{2012}}\)

\(\Rightarrow A=\left(1-\frac{1}{3^{2012}}\right).\frac{1}{2}\)

\(\Rightarrow A=\frac{1}{2}-\frac{1}{3^{2012}}\)

Vì \(\frac{1}{2}-\frac{1}{3^{2012}}< \frac{1}{2}\) nên \(A< \frac{1}{2}\)

Vậy \(A< \frac{1}{2}\)

a) \(\frac{2^{10}+1}{2^{10}-1}\)và \(\frac{2^{10}-1}{2^{10}-3}\)

Ta có chính chất phân số trung gian là \(\frac{2^{10}+1}{2^{10}-3}\)

\(\frac{2^{10}+1}{2^{10}-1}>\frac{2^{10}+1}{2^{10}-3}\) ; \(\frac{2^{10}-1}{2^{10}-3}< \frac{2^{10}+1}{2^{10}-3}\)

Vì \(\frac{2^{10}+1}{2^{10}-1}>\frac{2^{10}+1}{2^{10}-3}>\frac{2^{10}-1}{2^{10}-3}\)

Nên \(\frac{2^{10}+1}{2^{10}-1}>\frac{2^{10}-1}{2^{10}-3}\)

b) \(A=\frac{2011}{2012}+\frac{2012}{2013}\)và \(B=\frac{2011+2012}{2012+2013}\)

Ta có : \(A=\frac{2011}{2012}+\frac{2012}{2013}>\frac{2011}{2013}+\frac{2012}{2013}=\frac{2011+2012}{2013}>\frac{2011+2012}{2012+2013}=B\)

Vậy A > B 

Có gì  sai cho sorry

a,

\(\frac{2^{10}+1}{2^{10}-1}=1+\frac{2}{2^{10}-1}< 1+\frac{2}{2^{10}-3}=\frac{2^{10}-1}{2^{10}-3}\)

b,

\(\frac{2011}{2012}+\frac{2012}{2013}>\frac{2011}{2012+2013}+\frac{2012}{2012+2013}=\frac{2011+2012}{2012+2013}\)