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Showing the sum of convergent and divergent sequence is divergent 
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#1
Mar3112, 11:34 AM

P: 78

1. The problem statement, all variables and given/known data
Show that the sum of a convrgent sequence and a divergent sequence must be a divergent sequence. What can you say about the sum of two divergent sequences? 2. Relevant equations A theorem in the book states: Let {a_n} converge to a and {b_n} converge to b, then the sequence {a_n+b_n} converges to a+b Definition of convergent: A sequence {a_n} converges to a number L if for each epsilon > 0 there exists a positive integer N such that a_nL < epsilon for all n ≥ N. The number L is called the limit of the sequence. The sequence {a_n} converges iuf there exists a number L such that {a_n} converges to L. The sequence {x_n} diverges if it does not converge. 3. The attempt at a solution I think I made this too simple and overlooked something... Here is my attempt. Let {x_n} be a sequence that converges to x. Let {y_n} be a divergent sequence. Suppose {x_n+y_n} is a convergent sequence. By the theorem in the book, this implies that {y_n} converges, but {y_n} diverges. Hence, the sum must be divergent. What can you say about the sum of two divergent sequences? Their sum is divergent. 


#2
Mar3112, 11:37 AM

Mentor
P: 18,019




#3
Mar3112, 11:42 AM

P: 78




#4
Mar3112, 11:43 AM

Mentor
P: 18,019

Showing the sum of convergent and divergent sequence is divergent



#5
Mar3112, 11:44 AM

P: 78




#6
Mar3112, 11:56 AM

P: 445




#7
Mar3112, 12:12 PM

P: 78

Would it be safe to put both sequences on top of each other? Since they are sequences, they are in 11 correspondence with the natural numbers. So {x_n+y_n}=x_1,y_1,...,x_n,y_n,...
If {x_n} converges to x, but {y_n} converges to nothing, the {y_n} terms are what break the sequence from converging. 


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