Friday, 4 October 2013

real analysis - Counterexample: If suminftyn=1an=1 and each angeq0, then limlimitsntonan=0



I'm studying for an exam and I ran into these problems. I'm having a feeling that this is not true. Hence, I don't need to prove. I need to just provide a counterexample. However, the appropriate example is not just coming. Any help?



Prove or give a counterexample:




  1. If n=1an=1 and each an0, then limnnan=0



  2. If an0 and n=1an converges, then limnnan=0



Answer



Let an2=1n2 and an=0 otherwise. Then nan does not tend to 0, and the series is convergent.



The sum is π26, but note you can multiply each term by the same constant to make the limit any nonzero number you wish.






Note, however, that if an is also nonincreasing, the situation is different: Series converges implies limnan=0



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