Sequence

MFB

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Can anyone help me?
 

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It would help to know what you have been learning.

But I suspect they want you to think about what [MATH]a_n[/MATH] would be if [MATH]a_{n+1} = a_n[/MATH]. How would that relate to a possible limit?

Once you have a candidate for the limit, you can try proving it.

EDIT: afos's idea is essentially equivalent to this, and a little more formal.
 
The sequence has the property that an=an12+1an1\displaystyle a_n= \frac{a_{n-1}}{2}+ \frac{1}{a_{n-1}}. Assuming the sequence has a limit, call it A. Of course, {an}\displaystyle \{a_n\} and {an1}\displaystyle \{a_{n-1}\} are just different ways of numbering that same sequence so limnan=A\displaystyle \lim_{n\to\infty} a_n= A and limnan1=A\displaystyle \lim_{n\to\infty} a_{n-1}= A. Taking the limit of both sides of an=an12+1an1\displaystyle a_n= \frac{a_{n-1}}{2}+ \frac{1}{a_{n-1}}, as afos suggested, gives A=A2+1A\displaystyle A= \frac{A}{2}+ \frac{1}{A} (as long as A0\displaystyle A\ne 0). Solve that for A. Once you have a value for A, try showing that this is a decreasing sequence having A (or a number less than A) as a lower bound.
 
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