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• Covered today: Power series
• For next time :

1. Find the sum of a given series or show that the series diverges:
   1. $\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\cdots$
2. 
   

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

   
   

3. 
   

   $$\sum_{n=1}^{\infty}\frac{1}{(2n-1)(2n+1)}$$

   
   

Decide whether the statement is true or false, if it is true prove it if false give a counterexample:

1. If $\sum a_{n}$ converge, the $\sum \frac{1}{a_{n}}$ diverge.
2. If $\sum a_{n}$ and $\sum b_{n}$ both diverge, then so does $\sum (a_{n}+b_{n})$.
3. If $a_{n} \ge c > 0$ for every $n$, then $\sum a_{n}$ diverge.
4. If $a_{n}>0$ and $\sum a_{n}$ converge, then $\sum a^{2}_{n}$ converge.

Determine the intervals of convergence of the power series:

1. 
   

   $$\sum_{n=0}^{\infty}\frac{x^{2n}}{\sqrt{n+1}}$$

   
   

2. 
   

   $$\sum_{n=0}^{\infty}\frac{1}{n}\left( \frac{x+2}{2}\right)^{n}$$

   
   

3. 
   

   $$\sum_{n=0}^{\infty}\frac{e^{n}}{n^{3}}(4-x)^{n}$$

   
   

Determine the power series representation of the function. On what interval each representation valid?

1. $\frac{1}{2-x}$ in powers of $x$.
2. $\ln(2-x)$ in powers of $x$.
3. $\frac{1}{(2-x)^{2}}$

Review Taylor series