Question 11

Determine with proof, whether or not each of the following series converges and if it converges, decide, with proof, whether or not the convergence is absolute.
Also, if a series converges, either determine its sum exactly, or give an estimate of the sum, with proof, accurate to within an error of at most $${\frac{1}{4}}$$:

• $${\mathcal{A} = \sum_{n = 0}^\infty \left(\frac{(-5)^n}{3^{2n}}\right)}$$
  
  
  
  
  
  
  
  
  
  
  
  
  
  
• $${\mathcal{B} = \sum_{n = 1}^\infty \left((-1)^n \sqrt{ \frac{n}{4n^2 + 1}}\right)}$$
  
  
  
  
  
  
  
  
  
  
  
  
  
  
• $${\mathcal{C} = \sum_{n = 1}^\infty \left(\frac{2^{n^2}}{n^n}\right)}$$