Convergence of Sequence Summation and Limit Prove

In summary, the problem asks to prove that the limit of a sequence ((A1 + 2A2 + ... + nAn)/n) is equal to 0 given that the sequence (An) is in R and the summation of (An) from n=1 to infinity is convergent. Using this information, we can rewrite the limit expression as the limit of the summation of (nAn)/n, and since the summation of (An) converges, we can conclude that the limit of (An) is equal to 0. We can then use the given hint to show that if (Bn) is a sequence where |Bn| < |An|, then the summation of (
  • #1
l888l888l888
50
0

Homework Statement


let (An) be a sequence in R with |summation from n=1 to infinity(An)|< infinity. Prove lim as n goes to infinity of ((A1 +2A2+...+nAn)/n) = 0


Homework Equations





The Attempt at a Solution


I think |summation from n=1 to infinity(An)|< infinity means the summation converges .I rewrote " lim as n goes to infinity of ((A1 +2A2+...+nAn)/n) = 0 " as "lim as n goes to infinity of ((summation from k=1 to n of nAn)/n)=0. Since I assmumed "summation from n=1 to infinity(An)" converges, that would imply the lim of An is 0. I don't know how to use this info to prove what i need to prove. Hope you can understand this. I don't know how to use tek.
 
Physics news on Phys.org
  • #2
Hint: If the summation of An converges and for all n ,Bn<An, then the summation of Bn converges.
 
  • #3
Forgive my missing absolute. |Bn|<|An|
 

1. What is the definition of convergence of a sequence?

The convergence of a sequence is the behavior of the terms in the sequence as the number of terms increases. A sequence is said to converge if the terms approach a single fixed value as the number of terms increases, otherwise it is said to diverge.

2. How do you prove the convergence of a sequence?

To prove the convergence of a sequence, one must show that the terms in the sequence get closer and closer to a specific value as the number of terms increases. This can be done by using mathematical techniques such as the limit definition, comparison test, or the ratio and root tests, among others.

3. What is the difference between a convergent and a divergent sequence?

A convergent sequence approaches a single fixed value as the number of terms increases, while a divergent sequence does not have a single fixed value and may either approach infinity or oscillate between different values as the number of terms increases.

4. Can a sequence converge to more than one value?

No, a sequence can only converge to a single value. If a sequence has multiple limits, it is considered to be divergent.

5. How does the convergence of a sequence relate to the convergence of a series?

The convergence of a sequence is closely related to the convergence of a series. A series is the sum of all the terms in a sequence, and it converges if and only if the sequence of partial sums converges. This means that if a sequence converges, then the series will also converge, and if a sequence diverges, then the series will also diverge.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
937
  • Calculus and Beyond Homework Help
Replies
2
Views
658
  • Calculus and Beyond Homework Help
Replies
4
Views
866
  • Calculus and Beyond Homework Help
Replies
1
Views
766
  • Calculus and Beyond Homework Help
Replies
4
Views
849
  • Calculus and Beyond Homework Help
Replies
13
Views
899
  • Calculus and Beyond Homework Help
Replies
11
Views
1K
  • Calculus and Beyond Homework Help
Replies
13
Views
635
  • Calculus and Beyond Homework Help
Replies
23
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
767
Back
Top