The expression 22-1-22-1-32-1-32-1-42-1-42-1-20112-1-20112-1 lies in the interval

∑ #160; 1 lt;∑ #160; (n+1) ^ 2 +1 (n+1) ^ 2 1 lt;∑ #160; ( 1+ 2 n(n+1) #160; ) 2010 lt;∑ #160; #160; _ i=1 ^ 2010 (n+1) ^ 2 ...

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The expression $\frac{2^{2} + 1}{2^{2} - 1} + \frac{3^{2} + 1}{3^{2} - 1} + \frac{4^{2} + 1}{4^{2} - 1} + . . . . . . . . . . + \frac{(2011)^{2} + 1}{(2011)^{2} - 1}$ lies in the interval

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\frac{2^{2} + 1}{2^{2} - 1} + \frac{3^{2} + 1}{3^{2} - 1} + \frac{4^{2} + 1}{4^{2} - 1} + . . . . . . . . . . . + \frac{(2011)^{2} + 1}{(2011)^{2} - 1}

\frac{2^{2} + 1}{2^{2} - 1} + \frac{3^{2} + 1}{3^{2} - 1} + \frac{4^{2} + 1}{4^{2} - 1} + . . . . . + \frac{(2011)^{2} + 1}{(2011)^{2} - 1}

x = \frac{1}{1^{2}} + \frac{1}{3^{2}} + \frac{1}{5^{2}} + . . . .,

y = \frac{1}{1^{2}} + \frac{3}{2^{2}} + \frac{1}{3^{2}} + \frac{3}{4^{2}} + . .

z = \frac{1}{1^{2}} - \frac{1}{2^{2}} + \frac{1}{3^{2}} - \frac{1}{4^{2}} + . . . .,

\sqrt{1 + \frac{1}{1^{2}} + \frac{1}{2^{2}}} + \sqrt{1 + \frac{1}{2^{2}} + \frac{1}{3^{2}}} + \sqrt{1 + \frac{1}{3^{2}} + \frac{1}{4^{2}}} + . . . . n terms

\frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + . . . . u p t o \infty = \frac{π^{2}}{6}

1 - \frac{1}{2^{2}} + \frac{1}{3^{2}} - \frac{1}{4^{2}} + . . . . u p t o \infty

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