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- Use the arithmetic properties of convergent sequences to find the limits of the following sequences.

(a) ( 2*n*/(*n*^{2}+ 1) )

(b) ( (*n*^{2}- 2*n*+ 1)/(*n*^{2}+ 2*n*+ 1) )

(c) ( 3^{n}/(2^{n}+ 3^{n}) )

(d) (√(*n*+ 1) - √*n*) [Multiply top and bottom by √(*n*+ 1) + √*n*] - By looking for suitable divergent subsequences, prove that the following sequences are divergent.

(a) ( sin(*n*π/3) )

(b) ( (-1)^{n}*n*/(2*n*+1) ) - Prove that
*n*!/*n*^{n}<^{1}/_{n}, and hence show that the sequence*n*!/*n*^{n}converges to 0.

[The Scottish mathematician (who was also the manager of a mine at Leadhills) James Stirling (1692 to 1770) showed that the sequence (*n*!/(*e*^{n}*n*^{n+1/2}) ) converges to √(2π).] - A sequence (
*a*_{n}) satisfies*a*_{n+1}= 2*a*_{n}/(1+*a*_{n}) for*n*= 1, 2, ...

(a) If*a*_{1}= 2, prove that the sequence is monotonic decreasing and bounded and find its limit.

(b) If*a*_{1}=^{1}/_{2}, prove that the sequence is monotonic increasing and bounded and find its limit. - Let (
*a*_{n}) be a sequence such that the subsequence of even terms (*a*_{2n}) and the subsequence of odd terms (*a*_{2n-1}) both converge to the same limit*α*. Prove that (*a*_{n}) converges to*α*. - Let (
*f*_{n}) be the Fibonacci sequence (1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144 , ... ) satisfying*f*_{1}=*f*_{2}= 1 and*f*_{n+1}=*f*_{n}+*f*_{n-1}.

Let (*r*_{n}) be the sequence of ratios of successive Fibonacci numbers. So*r*_{n}=*f*_{n+1}/*f*_{n}and (*r*_{n}) = (1 , 2 ,^{3}/_{2},^{5}/_{3},^{8}/_{5},^{13}/_{8},^{21}/_{13}, ...) . Is this sequence monotonic? [Use a calculator.]

Prove that*r*_{n+1}= 1 + 1/*r*_{n}and hence prove that*if*this sequence were convergent then its limit would be (√5 + 1)/2. (This is the number that the Ancient Greeks called the*Golden Ratio*.)

Prove that*r*_{n+2}= (2*r*_{n}+ 1)/(*r*_{n}+ 1).

Hence show that the subsequence of odd terms and the subsequence of even terms are monotonic and bounded.

Deduce that (*r*_{n}) is convergent to the Golden Ratio.

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