Solution:
Dạng tổng quát :
\(\sqrt{1+k^2+\frac{k^2}{\left(k-1\right)^2}}=\sqrt{\frac{\left(1+k^2\right)\left(k+1\right)^2+k^2}{\left(k-1\right)^2}}\)
\(=\sqrt{\frac{k^4-2k^3+3k^2-2k+1}{\left(k-1\right)^2}}=\sqrt{\frac{\left(k^2-k\right)^2+2\left(k^2-k\right)+1}{\left(k-1\right)^2}}\)
\(=\sqrt{\frac{\left(k^2-k+1\right)^2}{\left(k-1\right)^2}}=\frac{k^2-k+1}{k-1}\)
\(=\frac{k\left(k-1\right)+1}{k-1}=k+\frac{1}{k-1}\)
Áp dụng ta có :
\(S=\sqrt{1+2020^2+\frac{2020^2}{2019^2}}-\frac{2020}{2019}\)
\(S=2020+\frac{1}{2019}-\frac{2020}{2019}\)
\(S=2020+\frac{-2019}{2019}\)
\(S=2020-1\)
\(S=2019\)
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