So I have a problem and have simplified the part I am confused about below.
If ∑∞m=1c<∞ and 0≤c≤1, then limn→∞∑∞m=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=limn→∞∑ni=1c=limn→∞nc=∞
If c=0 then ∑∞i=1c=0
If c<0then ∑∞i=1c=−∞
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