Summation Using Sigma

Q. $\text{Let}{S}_n={\sum }_{l=1}^n\left(\frac{l^4+l^{3}n+l^2{n}^2+2n^4}{n^5}\right)\text{and}{T}_n={\sum }_{l=0}^{n-1}\left(\frac{l^4+l^{3}n+l^2{n}^2+2n^4}{n^5}\right),(n=1,2,3,...)\text{then}$

1. $T_n>\frac{167}{60}$
2. $T_n<\frac{167}{60}$
3. $S_n>\frac{167}{60}$
4. $S_n<\frac{167}{60}$

Q. The value of $\underset{n\to \infty }{lim}\underset{k=1}{\sum ^n}(\frac{n-k}{n^2})\cos \frac{4k}{n}=\frac{1}{a}(1-\cos c),$ then

1. $a+c=20$
2. $\frac{a}{c}=4$
3. $a-c=12$
4. $ac=16$

Q.

If the value of ${lim}_{n\to \infty }\left(n^{-3/2}\right){\sum }_{j=1}^{6n}\sqrt{j}$ is equal to $\sqrt{N}$, then value of N/12 is___

Q. The value of $\underset{n\to \infty }{lim}\frac{\left[\right(n+1\left)\right(n+2)...(n+n){]}^{\frac{1}{n}}}{n}$ is:

1. $\frac{4}{e^2}$
2. $\frac{3}{e}$
3. $\frac{3}{e^2}$
4. $\frac{4}{e}$