Proof of Sequence Convergence: Find Sum of Sequence

In summary, the conversation discusses proving the convergence of a sequence with a starting term of 5 and the following terms calculated using a recurrence relation. The speaker suggests using induction to prove that the sequence is decreasing and has a lower bound of 3. They also mention finding the limit once convergence is proven.
  • #1
kryckmeister
1
0
Sequence

I'm really crappy at this, but I'm supposed to prove that the sequence:
a1=5 and the rest from a(n+1) = 0.5 * ( a(n) + 10/(a(n) ) is convergent.
And also find its sum. HELP me with this please.
 
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  • #2
The first thing I would do is calculate a few terms: a1= 5, a2= 0.5(5+ 10/5)= 3.5, a3= 0.5(3.5+ 10/3.5))= 3.1785... . Hmm, looks like it is a decreasing sequence! If I could prove that (perhaps by induction) and prove that is has a lower bound (which is obvious: every number is positive) then, by the "monotone convergence property", the sequence converges.
It is clear that a2< a1. Assume that ak+1[/sup]< ak[/sup] for some k. Can you prove, using the recurrance relation, that then ak+2< ak?

(It occurs to me that, in order to deal with the "10/an" part, in which an is in the denominator, you will need a lower bound, other than 0, on the sequence. Do you see how to prove that, in fact, 3 is a lower bound on the sequence? Use induction again.)

By the way, once you have proved that the sequence converges, it is easy to actually find the limit: If the sequence converges to a, then, taking the limit on both sides of a(n+1)= 0.5*(a(n)+ 10/a(n)) you get lim(a(n+1))= 0.5 (lim(a(n))+ 10/lim(a(n))) which, since all of those limits are the same, is a= 0.5(a+ 10/a). Solve that equation for a.
 
  • #3
Thread from General Math merged here. Welcome to the PF, kryckmeister. Please take care to post homework/coursework questions here in the Homework Help forums, and not in the general technical forums. Thanks!
 

FAQ: Proof of Sequence Convergence: Find Sum of Sequence

What is a sequence?

A sequence is a list of numbers that follow a certain pattern or rule. Each number in the sequence is called a term.

What is proof of sequence convergence?

Proof of sequence convergence is a mathematical method used to determine if a sequence approaches a particular value, also known as the limit, as the number of terms in the sequence increases.

How do you determine the sum of a sequence?

The sum of a sequence is found by adding up all the terms in the sequence. This can be done by manually adding the terms or by using a formula specific to the type of sequence.

What is the importance of finding the sum of a sequence?

Finding the sum of a sequence can help in understanding the behavior and patterns of the sequence. It can also be used in various real-life applications such as finance, physics, and computer science.

What are some common methods used to prove sequence convergence?

Some common methods used to prove sequence convergence include the Squeeze Theorem, the Monotone Convergence Theorem, and the Cauchy Criterion. These methods involve comparing the given sequence to a known convergent sequence or using specific criteria to determine convergence.

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