Sunday, December 17, 2017

summation - Convince me: limit of sum of a constant is infinity



So I have a problem and have simplified the part I am confused about below.



If m=1c< and 0c1, then limnm=nc=0 which implies c=0.




My general intuition says that because the sum of infinitely many non-negative c's is less than infinity, than c=0 because the sum of an infinitely many positive numbers will always be infinity.



The limit is where I am confused. I feel like the limit will always be 0 even if c>0. It also feels like the limit is not necessary to show c=0.


Answer



If c>0 then i=1c=limnni=1c=limnnc=



If c=0 then i=1c=0



If c<0then i=1c=



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