Ta có:

\(\left(2015^{2015}+2016^{2015}\right)^{2016}=\left(2015^{2015}+2016^{2015}\right)^{2015}.\left(2015^{2015}+2016^{2015}\right)\)

\(>\left(2015^{2015}+2016^{2015}\right)^{2015}.2016^{2015}=\left[\left(2015^{2015}+2016^{2015}\right)2016\right]^{2015}\)

\(>\left(2015^{2015}.2015+2016^{2015}.2016\right)^{2015}=\left(2015^{2016}+2016^{2016}\right)^{2015}\)

Vậy \(\left(2015^{2015}+2016^{2015}\right)^{2016}>\left(2015^{2016}+2016^{2016}\right)^{2015}\)

1. Ta sẽ chứng minh \(2015^{2016}>2016^{2015}\)

\(\Leftrightarrow2016^{2015}-2015^{2016}< 0\Leftrightarrow2016^{2016}-2016.2015^{2016}< 0\)

\(\Leftrightarrow2016.2016^{2016}-2015.2016^{2016}-2016.2015^{2016}< 0\)

\(\Leftrightarrow2016\left(2016^{2016}-2015^{2016}\right)< 2015.2016^{2016}\)

\(\Leftrightarrow2016\left(2016^{2015}+2016^{2014}.2015+...+2015^{2015}\right)< 2015.2016^{2016}\)

\(\Leftrightarrow2016^{2015}.2015+...+2016.2015^{2015}< 2014.2016^{2016}\)

\(\Leftrightarrow2016^{2014}.2015+2016^{2013}.2015^2+...+2015^{2015}< 2014.2016^{2015}\)

\(\Leftrightarrow2015^{2015}< \left(2016^{2015}-2015.2016^{2014}\right)+\left(2016^{2015}-2015^2.2016^{2013}\right)\)

\(+...+\left(2016^{2015}-2015^{2014}.2016\right)\)

\(\Leftrightarrow2015^{2015}< 2014.2016^{2014}+2013.2016^{2014}.2015+...+2016.2015^{2013}\)

Lại có \(2015^{2015}=2014.2015^{2014}+2015^{2014}< 2014.2016^{2014}+2015^{2014}\)

Mà \(2015^{2014}< 2013.2016^{2014}.2015\)

nên \(2015^{2014}< 2014.2016^{2014}+2013.2016^{2014}.2015+...+2016.2015^{2013}\)

Vậy \(2015^{2016}>2016^{2015}.\)

\(3^{2016}=3.3^{2015}\)

Ta có : \(3^{2015}=3^{2015}\)

Mà : \(4>3\)

\(\Rightarrow3.3^{2015}< 4.3^{2015}\)

Vậy : \(3^{2016}< 4.3^{2015}\)

Ta có : \(3^{2016}=3.3^{2015}\)

Khi đó ta có \(3.3^{2015}\) so sánh với \(4.3^{2015}\)

Vì 3 < 4

=> \(3^{2016}< 4.3^{2015}\)

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!!! :)))

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

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

\(A=2017\)

\(A=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{2015}{2016}\)

\(\Rightarrow A>\frac{1}{2}.\frac{2}{3}.\frac{3}{4}...\frac{2013}{2014}\)

\(\Rightarrow A>\frac{1.2.3...2013}{2.3.4...2014}\)

\(\Rightarrow A>\frac{1}{2014}>\frac{1}{2017}\)

Vậy \(A>\frac{1}{2017}\left(đpcm\right)\)

A lớn hơn 4

B bé hơn 3

A = 15/14 + 16/15 + 17/16 + 18/17

Ta thấy :

15/14 > 1

16/15 > 1

17/16 > 1

18/17 > 1

=> A > 4

B tương tự

Ta có:

A =\(\frac{15}{14}< 4\)\(;\)\(\frac{16}{15}< 4\)\(;\)\(\frac{17}{16}< 4\)\(;\)\(\frac{18}{17}< 4\)

Vậy A bé hơn 4.

B =\(\frac{2015}{2016}< 3\)\(;\)\(\frac{2016}{2017}< 3\)\(;\)\(\frac{2016}{2019}< 3.\)

Vậy B bé hơn 3.

uk, cái bạn tên Phong Thần công nhận giỏi thật nha

A > B

\(y=\frac{1}{x^2+\sqrt{x}}\)