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Xét với n là số nguyên thì : \(\frac{1}{2^{-n}+1}+\frac{1}{2^n+1}=\frac{1}{\frac{1}{2^n}+1}+\frac{1}{2^n+1}=\frac{2^n}{2^n+1}+\frac{1}{2^n+1}=\frac{2^n+1}{2^n+1}=1\)
Vậy ta nhóm hợp lí như sau :
\(S=\left(\frac{1}{2^{-2013}+1}+\frac{1}{2^{2013}+1}\right)+\left(\frac{1}{2^{-2012}+1}+\frac{1}{2^{2012}+1}\right)+...+\left(\frac{1}{2^{-1}+1}+\frac{1}{2^1+1}\right)+\frac{1}{2^0+1}\)
\(=1+1+...+1+\frac{1}{2}\) (2013 số hạng 1)
\(=2013+\frac{1}{2}\)
= \(\frac{1}{\sqrt{2}\left(\sqrt{2}+1\right)}+\frac{1}{\sqrt{6}\left(\sqrt{3}+\sqrt{2}\right)}+...+\frac{1}{\sqrt{2012}.\sqrt{2013}\left(\sqrt{2013}+\sqrt{2012}\right)}\)
= \(\frac{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}{\sqrt{2\left(\sqrt{2}+1\right)}}+...+\frac{\left(\sqrt{2013}-\sqrt{2012}\right)\left(\sqrt{2013}+\sqrt{2012}\right)}{\sqrt{2012}\sqrt{2013}\left(\sqrt{2012}+\sqrt{2013}\right)}\)
= \(\frac{\sqrt{2}-1}{\sqrt{2}}+...+\frac{\sqrt{2013}-\sqrt{2012}}{\sqrt{2012}\sqrt{2013}}\)
= \(1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2012}}-\frac{1}{\sqrt{2013}}\)
= \(\frac{\sqrt{2013}-1}{\sqrt{2013}}=\frac{2013-\sqrt{2013}}{2013}\)
Tổng quát:\(\sqrt{1+\frac{1}{a^2}+\frac{1}{\left(a+1\right)^2}}=\frac{a^2+a+1}{a\left(a+1\right)}=1+\frac{1}{a\left(a+1\right)}\)\(=1+\frac{1}{a}-\frac{1}{a+1}\)
Áp dụng vào lm thôi
Xét số hạng tổng quát: \(\frac{1}{\sqrt{n}+\sqrt{n+1}}=\frac{\sqrt{n+1}-\sqrt{n}}{\left(\sqrt{n+1}+\sqrt{n}\right)\left(\sqrt{n+1}-\sqrt{n}\right)}=\sqrt{n+1}-\sqrt{n}\) (do \(\sqrt{n+1}-\sqrt{n}>0\forall n\in\mathbb{N}\text{ nên ta có thể nhân liên hợp}\))
Áp dụng vào và ta có:
\(VT=\sqrt{2}-1+\sqrt{3}-\sqrt{2}+...+\sqrt{2013^2}-\sqrt{2013^2-1}\)
\(=\sqrt{2013^2}-1=2013-1=2012^{\left(đpcm\right)}\)
NX \(\frac{1-\sqrt{n}+\sqrt{n+1}}{1+\sqrt{n}+\sqrt{n+1}}\) =\(\frac{\left(1-\sqrt{n}+\sqrt{n+1}\right)\left(\sqrt{n+1}-\sqrt{n}-1\right)}{\left(\sqrt{n+1}\right)^2-\left(\sqrt{n}+1\right)^2}\)
=\(\frac{\left(\left(\sqrt{n+1}-\sqrt{n}\right)^2-1^2\right)}{n+1-n-1-2\sqrt{n}}\) \(=\frac{n+1+n-2\sqrt{\left(n+1\right)n}-1}{-2\sqrt{n}}=\frac{2n-2\sqrt{n\left(n+1\right)}}{-2\sqrt{n}}\)
=\(\frac{n-\sqrt{n\left(n+1\right)}}{-\sqrt{n}}=\frac{n}{-\sqrt{n}}+\frac{\sqrt{n\left(n+1\right)}}{\sqrt{n}}=-\sqrt{n}+\sqrt{n+1}\)
thay vao Q ta co
Q= \(-\sqrt{2}+\sqrt{3}-\sqrt{3}+\sqrt{4}-...-\sqrt{2012}+\sqrt{2013}=-\sqrt{2}+\sqrt{2013}\)
a, \(S=\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+...+\frac{1}{2015.2017}\)
\(\Rightarrow\) \(2S=\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+...+\frac{2}{2015.2017}\)
\(\Rightarrow\) \(2S=1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{2015}-\frac{1}{2017}\)
\(\Rightarrow\) \(2S=1-\frac{1}{2017}\)
\(\Rightarrow\) \(2S=\frac{2016}{2017}\)
\(\Rightarrow\) \(S=\frac{1008}{2017}\)
Xét \(P=\sqrt{\frac{1}{1^2}+\frac{1}{a^2}+\frac{1}{\left(a+1\right)^2}}\) với a>0
\(P^2=1+\frac{1}{a^2}+\frac{1}{\left(a+1\right)^2}\)
\(=\frac{a^2\left(a+1\right)^2+\left(a+1\right)^2+a^2}{a^2\left(a+1\right)^2}\)
\(=\frac{a^2\left(a^2+2a+1+1\right)+\left(a+1\right)^2}{a^2\left(a+1\right)^2}\)
\(=\frac{a^4+2a^2\left(a+1\right)+\left(a+1\right)^2}{a^2\left(a+1\right)^2}\)
\(=\frac{\left(a^2+a+1\right)^2}{a^2\left(a+1\right)^2}\)
\(=\left(\frac{a^2+a+1}{a\left(a+1\right)}\right)^2\)
Do a>o nên \(P=\frac{a^2+a+1}{a\left(a+1\right)}=1+\frac{1}{a}-\frac{1}{a+1}\)
Áp dụng kết quả của P ta có:
\(A=\left(1+\frac{1}{1}-\frac{1}{2}\right)+\left(1+\frac{1}{2}+\frac{1}{3}\right)+....+\left(1+\frac{1}{2012}-\frac{1}{2013}\right)\) \(A=2012+\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-.....+\frac{1}{2012}-\frac{1}{2013}\right)\)
\(A=2012+1-\frac{1}{2013}\)
\(A=2013-\frac{1}{2013}=\frac{4052168}{2013}\)
Vậy \(A=\frac{4052168}{2013}\)
\(S=\frac{2^{2013}}{2^{2013}+1}+\frac{2^{2012}}{2^{2012}+1}+....+\frac{1}{2^{2012}+1}+\frac{1}{2^{2013}+1}\)
=(\(\frac{2^{2013}}{2^{2013}+1}+\frac{1}{2^{2013}+1}\))+(\(\frac{2^{2012}}{2^{2012}+1}+\frac{1}{2^{2012}+1}\))+...+ \(\frac{1}{2}\) ( có 2013 dấu ngoặc )
= 1+ 1+.....+ \(\frac{1}{2}\) = 2013\(\frac{1}{2}\)