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- Thread starter Kirono23
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Let:
[MATH]S_n=\sum_{k=1}^{n}\left(\frac{1}{\sqrt{k+1}+\sqrt{k}}\right)=\sum_{k=1}^{n}\left(\sqrt{k+1}-\sqrt{k}\right)[/MATH]
[MATH]S_n=\sum_{k=1}^{n}\left(\sqrt{k+1}\right)-\sum_{k=1}^{n}\left(\sqrt{k}\right)=\sum_{k=1}^{n}\left(\sqrt{k+1}\right)-\sum_{k=0}^{n-1}\left(\sqrt{k+1}\right)[/MATH]
[MATH]S_n=\sqrt{n+1}+\sum_{k=1}^{n-1}\left(\sqrt{k+1}\right)-\sum_{k=1}^{n-1}\left(\sqrt{k+1}\right)-1=\sqrt{n+1}-1[/MATH]
[MATH]S_n=\sum_{k=1}^{n}\left(\frac{1}{\sqrt{k+1}+\sqrt{k}}\right)=\sum_{k=1}^{n}\left(\sqrt{k+1}-\sqrt{k}\right)[/MATH]
[MATH]S_n=\sum_{k=1}^{n}\left(\sqrt{k+1}\right)-\sum_{k=1}^{n}\left(\sqrt{k}\right)=\sum_{k=1}^{n}\left(\sqrt{k+1}\right)-\sum_{k=0}^{n-1}\left(\sqrt{k+1}\right)[/MATH]
[MATH]S_n=\sqrt{n+1}+\sum_{k=1}^{n-1}\left(\sqrt{k+1}\right)-\sum_{k=1}^{n-1}\left(\sqrt{k+1}\right)-1=\sqrt{n+1}-1[/MATH]