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xinlei · Oct 6, 2009, 07:02 AM

Hi, can anyone help me with this questions? Thanks!!

lim xn n->∞ xn= {1/(n2 +1)} + {2/(n2 + 2)} +... + {n/(n2 + n)}

and

lim yn n->∞ yn= {1/sq(n2 + 1)} + {1/sq(n2 + 2)} +... + {1/sq(n2 + n)}

Thanks a lot! :)

ebaines · Oct 6, 2009, 10:23 AM

I can help with the first problem. Note that for large values of n:

$<br /> \frac 1 {n^2+1} + \frac 2 {n^2+2} + ... + \frac n {n^2+n} \ = \ \frac {1 + 2 + ...+ n} {n^2} \\<br /> = \frac {\sum _ {i=1} ^n i }{n^2} = \frac {n(n+1)} {2 n^2}<br />$

The limit of this as n goes to $\infty$ is 1/2.