Monday, 4 May 2015

sequences and series - can't determine the convergence/divergence here



Let tn=1n(1+12++1n), n=1,2, then I want to know if n=1tn converges/diverges and the sequence{tn} converges and diverges
for it I thought of finding lim but how to solve this limit I can do it if it is presented as a Riemann sum like if there is n in the denominator of r


Answer



First t_n>1/n , so \sum_{n=1}^\infty=\infty .



For t_n alone,

t_n =\frac1n\,\sum_{k=1}^n\frac1 {\sqrt k} =\left (\frac1n\,\sum_{k=1}^n\frac1 {\sqrt{ k/n}}\right)\,\frac1 {\sqrt n}.
The expression in brackets converges to \int_0^1\frac1 {\sqrt t}\,dt=2 , so the product converges to zero:\lim_{n\to\infty}t_n=\lim_{n\to\infty}\left (\frac1n\,\sum_{k=1}^n\frac1 {\sqrt{ k/n}}\right)\,\lim_{n\to\infty}\frac1 {\sqrt n}=2\times0=0.


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