Some series

A series is said to converge if has a finite limit as .

Exercises

1. Use Octave to make a conjecture about the convergence of
   
   
   as .
2. Use Octave to make a conjecture about the convergence of
   
   
   as .
3. 1. Rewrite the series
      
      using a Σ
   2. Use Octave to make a conjecture about the convergence of this series as
4. Iterate to find the sum of the first 50 terms of
   
   
   then multiply the result with 4. What happens when you iterate 500 times? 5000 times? 50000 times?