Exercises 5

1. Use the arithmetic properties of convergent sequences to find the limits of the following sequences.
   (a) ( 2n/(n²+ 1) )
   (b) ( (n²- 2n + 1)/(n²+ 2n + 1) )
   (c) ( 3ⁿ/(2ⁿ+ 3ⁿ) )
   (d) (√(n + 1) - √n ) [Multiply top and bottom by √(n + 1) + √n]

   Solution to question 1

2. By looking for suitable divergent subsequences, prove that the following sequences are divergent.
   (a) ( sin(nπ/3) )
   (b) ( (-1)ⁿn/(2n+1) )

   Solution to question 2

3. Prove that n!/nⁿ< ¹/_n, and hence show that the sequence n!/nⁿ 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!/(eⁿn^n+1/2) ) converges to √(2π).]

   Solution to question 3

4. A sequence (a_n) satisfies a_n+1= 2a_n/(1+a_n) for n = 1, 2, ...
   (a) If a₁= 2, prove that the sequence is monotonic decreasing and bounded and find its limit.
   (b) If a₁= ¹/₂ , prove that the sequence is monotonic increasing and bounded and find its limit.

   Solution to question 4

5. 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.

   Solution to question 5

6. Let (f_n) be the Fibonacci sequence (1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144 , ... ) satisfying f₁ = f₂ = 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 = (2r_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.

   Solution to question 6