31. infinite series - comparison tests

the convergence or divergence of $\sum_{n=1}^{\infty}\ \frac{2}{n^2-4}$ can be determined by  

• Direct Comparison to  $\sum_{n=1}^{\infty}\ \frac{2}{n^2}$  

• Limit Comparison to $\sum_{n=1}^{\infty}\ \frac{1}{n+2}$  

• Limit Comparison to  $\sum_{n=1}^{\infty}\ \frac{2}{n^2}$  

• Direct Comparison to the Harmonic Series

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

• diverges by Direct Comparison to the Harmonic Series 

• converges by Direct Comparison to  $\sum_{n=1}^{\infty}\ \frac{1}{n}$  

• converges by Direct Comparison to  $\sum_{n=1}^{\infty}\ \frac{1}{n^2}$ 

• diverges by Limit Comparison to  $\sum_{n=1}^{\infty}\ \frac{1}{n^2}$ 

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

• Direct Comparison to the Harmonic Series

• it is the Harmonic Series

• Limit Comparison to  $\sum_{n=1}^{\infty}\ \frac{1}{n^2}$  yields  $\infty$  

• Direct Comparison to a Geometric Series where  $\left|r\right|=2$  

convergence or divergence for  $\sum_{n=1}^{\infty}\ \frac{3^n+5}{2^n-1}$  can be determined by 

• Limit Comparison to  $\sum_{n=1}^{\infty}\left(\frac{3}{2}\right)^n$  

• Direct Comparison to  $\sum_{n=1}^{\infty}\ \frac{1}{2^n}$  

• GST since  $\left|r\right|=5$  

• p-series test since  $p=\frac{3}{2}$  

 $\sum_{n=1}^{\infty}\ \frac{n^2+1}{n^4-2n^2+1}$  converges because

•  $\lim_{n\rightarrow\infty}\ \frac{n^2+1}{n^4-2n^2+1}=0$  

•  $\lim_{n\rightarrow\infty}\ \frac{n^4+n^2}{n^4-2n^2+1}=1$  

•  $\lim_{n\rightarrow\infty}\ \frac{n^2+1}{n^4-2n^2+1}$  is finite and positive

• it's a p-series with  $p=4$  

which test is right for determining the convergence or divergence of  $\sum_{n=1}^{\infty}\ \frac{3}{n+4}$ 

• Direct Comparison to  $\sum_{n=1}^{\infty}\ \frac{1}{n}$  

• Direct Comparison to  $\sum_{n=1}^{\infty}\ \frac{1}{n^2}$  

• Limit Comparison to  $\sum_{n=1}^{\infty}\ \frac{1}{n}$  

• GST where  $r=\frac{3}{4}$  

the convergence of  $\sum_{n=1}^{\infty}\ \frac{2}{5^n+1}$ can be proven with the comparison series:  

•  $\sum_{n=1}^{\infty}\left(\frac{1}{5}\right)^n$  

•  $\sum_{n=1}^{\infty}\left(\frac{2}{5}\right)^n$  

•  $\sum_{n=1}^{\infty}\ 5^n$  

• the Harmonic Series

In comparison with the divergent series  $\sum_{ }^{ }b_n$  ,  $\lim_{n\rightarrow\infty}\ \frac{a_n}{b_n}=\frac{2}{3}$  

•  $\therefore\sum_{ }^{ }a_n$  diverges by Limit Comparison

•  $\therefore\sum_{ }^{ }a_n$  converges by Limit Comparison

• the Limit Comparison test is inconclusive

•  $\therefore\ \sum_{ }^{ }a_n=\frac{2}{3}$ 

In comparison with the convergent series  $\sum_{ }^{ }b_n$  ,  $a_n>b_n$  

•  $\therefore\sum_{ }^{ }a_n$  diverges by Direct Comparison

•  $\therefore\sum_{ }^{ }a_n$  converges by Direct Comparison

• the Direct Comparison test is inconclusive

•  $\therefore\ \sum_{ }^{ }a_n=\sum_{ }^{ }b_n$ 

The best infinite series test is

• Limit Comparison

• GST

• p-series

• Direct Comparison