Convergence Divergence Calculus Taylor

I

II

II and III

I and III

$$\sum_{n=1}^{\infty}\frac{n}{n^3+n^2}$$  

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

$$\sum_{n=1}^{\infty}\frac{1}{\sqrt{n^4+2n}}$$  

all of the above

$$\sum_{n=1}^{\infty}\frac{n^2+n}{3n^2+5}$$

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

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

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

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

$$\sum_{n=1}^{\infty}\frac{3^n}{5^n}$$

$$\sum_{n=1}^{\infty}\frac{5n^2}{3n^3+n^2}$$

$$\sum_{n=1}^{\infty}\frac{1}{\tan^{-1}n}$$

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

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

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

all of the above

$$\sum_{n=1}^{\infty}\frac{5^n}{n^2}$$

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

$$\sum_{n=1}^{\infty}\frac{5^n}{n!}$$

$$\sum_{n=1}^{\infty}\frac{n^2}{5^n}$$

Converges absolutely

Converges conditionally

Diverges

Cannot be determined

$$-1<p<1$$  

$$p>1$$  

$$p>0$$  

$$p<0$$  

The series converges if $$L=\frac{1}{2}$$  

The series converges if $$L=1$$  

The series converges if $$L=2$$  

The series converges if $$L=\infty$$  

I only

III only

I and II only

I and III only