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• Covered today: Taylor series
• For next time :

1. Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
   1. 
      

      $$\sum_{n=1}^{\infty}\frac{n^{2}}{2^{2}}$$

      
      

   2. 
      

      $$\sum_{n=1}^{\infty}\frac{(-1)^{n}}{\sqrt{n}}$$

      
      

   3. 
      

      $$\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n\sqrt{n}}$$

      
      

2. A series $\sum a_{n}$ is defined by the equations,
   $a_{1}=1$, $a_{n+1}=\frac{2+\cos(n)}{\sqrt{n}}a_{n}$
   Determine whether the series converge.
3. Show that the function

   
   

   $$J_{0}(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n}}{2^{2n}(n!)^{2}}$$

   
   

   Satisfy the differential equation
   

   $$x^{2}J^{\lq\lq }_{0}(x)+xJ^{\lq }_{0}(x)+x^{2}J_{0}(x)=0$$

   
   

4. Estimate the number of terms in Maclaurin series of $e^{x}$ that should be used to estimate $e^{0.1}$ to within $0.00001$

Read for next time: 3.1-3.3