Will the following converge or diverge? $\sum\limits_{n=1}^\infty \frac{ n^2 + 1 }{ n^5 + n + 1 } $

Calculate $\sum\limits_{n=1}^\infty \frac{1}{n^2+2n} $.

Calculate $\sum\limits_{n=1}^\infty \frac{1}{4n^2-1} $.

By use of the Squeeze Theorem calculate the limit $ \lim\limits_{n \to  \infty } \frac{1}{\sqrt{n^2+1}} + \frac{1}{\sqrt{n^2+2}}  + \frac{1}{\sqrt{n^2+n}} $.

Find the Maclaurin Series of $\frac{4x^2-3}{\left(1-x\right)\left(1-2x\right)^2}$ where $x$ is small i.e. less than $\frac{1}{2}$ in magnitude. But, you can’t use the formula for the Maclaurin Series however you may use the fact that $\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots $.  

By use of the Squeeze Theorem calculate the limit $ \lim\limits_{n \to  \infty } \frac{1}{n^2+1} + \frac{1}{n^2+2}  + \cdots + \frac{1}{n^2+n} $.  

Calculate the limit $ \lim\limits_{n \to  \infty } \frac{1}{ \sqrt{n+1}-\sqrt{n} } $.

Calculate the limit $ \lim\limits_{n \to  \infty } \frac{6n^2-n}{2n^2-5n+4} $.

Calculate the limit $ \lim\limits_{n \to  \infty } \frac{n^3-5n^2}{n^2-2n+1} $.