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 limntn=limn1nnr=11r 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 tn>1/n, so n=1=.



For tn alone,

tn=1nnk=11k=(1nnk=11k/n)1n.
The expression in brackets converges to 101tdt=2, so the product converges to zero:limntn=limn(1nnk=11k/n)limn1n=2×0=0.


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