MathSquareRoo said:I'm not good at writing proofs. So far I have:
Let {an} converge to A. Given epsilon>0, there exists N>0 s.t. lan-Al<epsilon for all n>N.
So l((a_n)-A)l<epsilon for all n>n.
Thus, we can write lim n--->infinity (a_n-A)=0.
Then, I'm not sure how to prove the statement's converse. Can someone help?
MathSquareRoo said:I'm not sure how to write the reverse. Would I start with: lim n--->infinity(a_n -A)=0. So given epsilon>0, there exists N>0 s.t. la_n-Al<epsilon for all n>N.
I don't know if this is correct, and I don't know where to go after that.
Sequence convergence in advanced calculus refers to the behavior of a sequence of numbers approaching a specific limit as the number of terms in the sequence increases. In other words, as the sequence continues, the terms get closer and closer to a specific value, known as the limit, and eventually reach it.
In sequence convergence, we are looking at the behavior of individual terms in a sequence as the number of terms increases. In series convergence, we are looking at the behavior of the sum of all the terms in a sequence as the number of terms increases.
The three main types of sequence convergence are:
- Convergence to a finite limit: This occurs when the terms of the sequence approach a specific finite value as the number of terms increases.
- Divergence to infinity: This occurs when the terms of the sequence increase or decrease without bound as the number of terms increases.
- Oscillating convergence: This occurs when the terms of the sequence alternate between two values as the number of terms increases.
To prove that a sequence converges, you can use one of the following methods:
- The epsilon-delta method: This involves choosing a specific value of epsilon (a small positive number) and finding a corresponding value of delta (a positive number) such that the terms of the sequence are within epsilon of the limit for all values greater than delta.
- The Monotone Convergence Theorem: This theorem states that if a sequence is bounded and monotonically increasing or decreasing, then it must converge.
- The Cauchy Criterion: This criterion states that a sequence converges if and only if it satisfies the Cauchy condition, which states that for any positive number epsilon, there exists a positive integer N such that the difference between any two terms of the sequence with indices greater than or equal to N is less than epsilon.
No, a sequence can only converge to one limit. If a sequence converges to more than one limit, it is known as a divergent sequence. This is because the definition of convergence states that the terms of the sequence must approach a specific finite value, and a sequence cannot approach more than one specific value at the same time.