Thread: Problem 39
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Old November 1st, 2007, 07:41 AM
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Let x_n=H_n(a,b), \ y_n=H_n(c,d).
(y_n) is ascending and unbounded.
By Stolz-Cesaro, we have
\displaystyle\lim_{n\to\infty}\frac{x_{n+1}-x_n}{y_{n+1}-y_n}=\lim_{n\to\infty}\frac{\frac{1}{a(n+1)+b}}{\frac{1}{c(n+1)+d}}=\lim_{n\to\infty}\frac{cn+c+d}{an+a+b}=\frac{c}{a}.

Then \displaystyle\lim_{n\to\infty}\frac{x_n}{y_n}=\frac{c}{a}
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