so sánh
P=1/1^2 + 1/2^2 +1/3^2 +...+1/2013^2 +1/2014^2 và Q=1+3/4
Những câu hỏi liên quan
So sánh:
P=1/1^2+1/2^2=1/3^2+1/4^2+...+1/2013^2+1/2014^2 và Q=1/3/4
So sánh 1/2^2 + 1/3^2 + 1/4^2 + ...... + 1/2013^2 và 2014/2013
ta có :\(\frac{1}{2^2}< \frac{1}{1.2}\)
\(\frac{1}{3^2}< \frac{1}{2.3}\)
\(\frac{1}{4^2}< \frac{1}{3.4}\)
\(............\)
\(\frac{1}{2013^2}< \frac{1}{2012.2013}\)
cộng vế với vế ta được :
\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2013^2}< 1-\frac{1}{2013}=\frac{2012}{2013}< \frac{2014}{2013}\)
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So sánh \(P=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+....+\frac{1}{2013^2}+\frac{1}{2014^2}\)và \(Q=1\frac{3}{4}\)
P=1/1^2+1/2^2+1/3^2+1/4^2+.......+1/2013^2+1/2014^2
Q=1+3/4
So sanh P va Q
so sánh 1/2^2+1/3^2+1/4^2+...+1/2013^2 và 2014/2013
Giúp mik với
Ta có: \(\frac{1}{2^2}< \frac{1}{1.2}=1-\frac{1}{2}\)
\(\frac{1}{3^2}< \frac{1}{2.3}=\frac{1}{2}-\frac{1}{3}\)
............
\(\frac{1}{2013^2}< \frac{1}{2012.2013}=\frac{1}{2012}-\frac{1}{2013}\)
=> \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2013^2}< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2012}-\frac{1}{2013}=1-\frac{1}{2013}< 1\)
=> \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2013^2}< 1\)
Mà \(\frac{2014}{2013}>1\)
=> \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2013^2}< \frac{2014}{2013}\)
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So sánh: \(P=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2013^2}+\frac{1}{2014^2}\)và \(Q=1\frac{3}{4}\)
25 tháng 6 2023 lúc 8:53
Hãy so sánh M và N biết:
M 1 2014 + 2 2013 + 3 2012 + ... + 2014 1
N 1 + (1 + 2) + (1 + 2 + 3) + ... + (1 + 2 + 3 + ... + 2014)
Đọc tiếp
Hãy so sánh M và N biết:
M = 1 2014 + 2 2013 + 3 2012 + ... + 2014 1
N = 1 + (1 + 2) + (1 + 2 + 3) + ... + (1 + 2 + 3 + ... + 2014)
Ta có: \(M=1\cdot2014+2\cdot2013+\cdots+2014\cdot1\)
\(=2\left(1\times2014+2\times2013+\cdots+1007\times1008\right)\)
\(=2\left\lbrack1\times\left(2015-1\right)+2\times\left(2015-2\right)+\cdots+1007\times\left(2015-1007\right)\right\rbrack\)
\(=2\cdot\left\lbrack2015\times\left(1+2+\cdots+1007\right)-\left(1^2+2^2+\cdots+1007^2\right)\right\rbrack\)
\(=2\cdot\left\lbrack2015\times1007\times\frac{1008}{2}-\frac{1007\times\left(1007+1\right)\times\left(2\times1007+1\right)}{6}\right\rbrack\)
\(=2\cdot\left\lbrack2015\times1007\times504-1007\times168\times2015\right\rbrack=2\times2015\times1007\times168\left(3-1\right)=4\times168\times2015\times1007\)
\(N=1+\left(1+2\right)+\cdots+\left(1+2+\cdots+2014\right)\)
\(\) \(=\frac{1\times2}{2}+\frac{2\times3}{2}+\cdots+\frac{2014\times2015}{2}\)
\(=\frac12\times\left(1\times2+2\times3+\cdots+2014\times2015\right)\)
\(=\frac12\times\left\lbrack1\times\left(1+1\right)+2\times\left(2+1\right)+\cdots+2014\times\left(2014+1\right)\right\rbrack\)
\(=\frac12\times\left\lbrack\left(1\times1+2\times2+\cdots+2014\times2014\right)+\left(1+2+\cdots+2014\right)\right\rbrack\)
\(=\frac12\times\left\lbrack\frac{2014\times\left(2014+1\right)\times\left(2\times2014+1\right)}{6}+\frac{2014\times2015}{2}\right\rbrack\)
\(=\frac12\times\left\lbrack1007\times2015\times1343+1007\times2015\right\rbrack=\frac12\times1007\times2015\times\left(1343+1\right)\)
=1007x2015x672
=4x168x2015x1007
Do đó: M=N
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frac{frac{1}{2}+frac{1}{3}+......+frac{1}{2013}}{frac{2012}{1}+frac{2012}{2}+frac{2011}{3}+...+frac{1}{2013}}frac{frac{1}{2}+frac{1}{3}+..+frac{1}{2013}}{frac{2012}{1}+2+frac{2012}{2}+1+frac{2011}{3}+1+...+frac{1}{2013}+1-2014}frac{frac{1}{2}+frac{1}{3}+...+frac{1}{2013}}{frac{2014}{1}+frac{2014}{2}+...+frac{2014}{2013}-2014}frac{frac{1}{2}+frac{1}{3}+...+frac{1}{2013}}{2014left(1+frac{1}{2}+frac{1}{3}+...+frac{1}{2013}-1right)}frac{1}{2014}
Đọc tiếp
\(\frac{\frac{1}{2}+\frac{1}{3}+......+\frac{1}{2013}}{\frac{2012}{1}+\frac{2012}{2}+\frac{2011}{3}+...+\frac{1}{2013}}\)
=\(\frac{\frac{1}{2}+\frac{1}{3}+..+\frac{1}{2013}}{\frac{2012}{1}+2+\frac{2012}{2}+1+\frac{2011}{3}+1+...+\frac{1}{2013}+1-2014}\)
=\(\frac{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2013}}{\frac{2014}{1}+\frac{2014}{2}+...+\frac{2014}{2013}-2014}\)
=\(\frac{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2013}}{2014\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2013}-1\right)}\)
=\(\frac{1}{2014}\)
1) 1/2 + 1/3 + 1/4 + ... + 1/2013 + 1/2014
2) 2014 + 2013/2 + 2012/3 + 2011/4 + ... + 2/2013 + 1/2014
