A) \[\sqrt{2}{{(\sqrt{2}+1)}^{2}}\]
B) \[{{(\sqrt{2}+1)}^{2}}\]
C) \[5\sqrt{2}\]
D) \[3\sqrt{2}+\sqrt{5}\]
E) \[0\]
Correct Answer: A
Solution :
In the given series \[a=\frac{\sqrt{2}+1}{\sqrt{2}-1}\] and \[r=\frac{1}{2+\sqrt{2}}=\frac{2-\sqrt{2}}{2}\] \[\therefore \] \[{{S}_{\infty }}=\frac{a}{1-r}=\frac{\frac{\sqrt{2}+1}{\sqrt{2}-1}}{1-\frac{2-\sqrt{2}}{2}}\] \[=\frac{\frac{\sqrt{2}+1}{\sqrt{2}-1}}{\frac{\sqrt{2}}{2}}\] \[=\frac{2+\sqrt{2}}{\sqrt{2}-1}\times \frac{\sqrt{2}+1}{\sqrt{2}+1}=\sqrt{2}(3+2\sqrt{2})\] \[=\sqrt{2}{{(\sqrt{2}+1)}^{2}}\]
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