Questions on Convergence of Sequence {an}: Find Its Limits

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In summary, the sequence {an} is defined by the formula a1=1, an+1=\sqrt{1+an/2} for n=1,2,3,4,... The question asks to deduce that {an} converges and find its limit. To do so, we can see that the sequence is increasing by looking at a few terms. We can also use a theorem about convergence of increasing sequences. To find the limit, we can take the limit on both sides of the equation a_{n+1}= \sqrt{1+ a_n/2}.
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axe69
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The sequence {an} is defined by

a1=1,an+1=[tex]\sqrt{1+an/2}[/tex] ,n=1,2,3,4,...

(a) a^2n-2<0 , (b)a^2a+1-a^2n>0. deduce that{an} converges and find its limits?

please help me get the answer...
 
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Are the (a) and (b) [itex]a^{2n-2}[/itex] and [itex]a^{2n-1}- a^{2n}> 0[/itex]? If so, what is "a" and what do they have to do with the given sequence?

Have you never seen any sequences like this before?? You don't seem to have even tried anything. While looking at a few terms won't prove anything, it might help you see what's happening: [itex]a_1= 1[/itex], [itex]a_2= \sqrt{1+ 1/2}= \sqrt{3/2}[/itex], [itex]a_3= \sqrt{1+ \sqrt{3/2}/2}= \sqrt{(2+ \sqrt{3/2})/2}[/itex]. Does that look like it is increasing? Do you know any theorem about convergence of increasing sequences?

As for finding the limit, what happens if you take the limit on both sides of the equation [itex]a_{n+1}= \sqrt{1+ a_n/2}[/itex]?
 

FAQ: Questions on Convergence of Sequence {an}: Find Its Limits

What is the definition of convergence of a sequence?

The convergence of a sequence is a property where the terms of the sequence approach a specific value, known as the limit, as the sequence progresses to infinity.

How do you determine the limit of a sequence {an}?

To find the limit of a sequence, you can use different methods such as the squeeze theorem, the ratio test, or the root test. These methods involve evaluating the behavior of the sequence as n approaches infinity.

Can a sequence have more than one limit?

No, a sequence can only have one limit. If a sequence has more than one limit, it is considered to be divergent, meaning it does not approach a specific value as n approaches infinity.

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

A convergent sequence approaches a specific value, known as the limit, as n approaches infinity. A divergent sequence does not approach a specific value and may have multiple limits or no limit at all.

How do you prove that a sequence is convergent?

To prove that a sequence is convergent, you can show that the terms of the sequence get closer and closer to a specific value as n approaches infinity. This can be done through different methods such as the epsilon-delta proof or the Cauchy criterion.

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