Ta thấy
\(1+3=4=2^2\)
\(1+3+5=9=3^2\)
\(1+3+5+7=16=4^2\)
\(...\)
\(1+3+5+7+...+101=51^2\)
\(\Rightarrow S=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{51^2}\)
mà \(\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{51^2}=\dfrac{\pi^2}{6}\) (Tự chứng minh công thức\(\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{n^2}=\dfrac{\pi^2}{6}\left(n=51\right)\) là công thức Basel )
\(\Rightarrow S=\dfrac{\pi^2}{6}-1=\dfrac{\pi^2-6}{6}\approx\dfrac{10-6}{6}=\dfrac{2}{3}\)
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