Summation Using Sigma

Q. For positive integer $n$, define $f\left(n\right)=n+\frac{16+5n-3n^2}{4n+3n^2}+\frac{32+n-3n^2}{8n+3n^2}+\frac{48-3n-n^2}{12n+3n^2}+...+\frac{25n-7n^2}{7n^2}$
Then, the value of $\underset{n\to \infty }{lim}f\left(n\right)$ is eequal to

1. $4-\frac{3}{4}{\log }_e\left(\frac{7}{3}\right)$
2. $4-\frac{4}{3}{\log }_e\left(\frac{7}{3}\right)$
3. $3+\frac{3}{4}{\log }_{e}7$
4. $3+\frac{4}{3}{\log }_{e}7$

Q. If $x_1=3$ and $x_{n+1}=\sqrt{2+x_n},n\ge 1$, then $\underset{n\to \infty }{lim}{x}_n$ is​​​​​​​

Q. $\underset{n\to \infty }{lim}\sum _{r=1}^n\frac{r}{n^2+n+r}$ is equal to

1. $0$
2. $\frac{1}{3}$
3. $\frac{1}{2}$
4. $1$

Q. If $\underset{x\to 2}{lim}\frac{\sum _{r=1}^n{x}^r-\sum _{r=1}^n{2}^r}{x-2}=f\left(n\right),$ then which of the following is/are CORRECT?

1. $f\left(2020\right)=1999(2{)}^{2020}+1$
2. $f\left(2020\right)=505(2{)}^{2022}+1$
3. $f\left(6\right)=321$
4. $f\left(10\right)=9217$

Q. Value of $\underset{n\to \infty }{lim}\frac{1}{n^3}[1+3+6+\cdots +\frac{n(n+1)}{2}]$ is

1. $\frac{1}{24}$

2. $\frac{1}{18}$

3. $\frac{1}{12}$

4. $\frac{1}{6}$

Q. The value of $\underset{n\to \infty }{lim}{\left(\frac{n!}{(mn{)}^n}\right)}^{1/n},$ where $m\in N$ is equal to

1. $\frac{1}{em}$
2. $\frac{m}{e}$
3. $em$
4. $\frac{e}{m}$

Q. $\underset{n\to \infty }{lim}(\frac{(n+1{)}^{1/3}}{n^{4/3}}+\frac{(n+2{)}^{1/3}}{n^{4/3}}+\cdots +\frac{(2n{)}^{1/3}}{n^{4/3}})$ is equal to

1. $\frac{3}{4}(2{)}^{4/3}-\frac{3}{4}$
2. $\frac{4}{3}(2{)}^{4/3}$
3. $\frac{3}{4}(2{)}^{4/3}-\frac{4}{3}$
4. $\frac{4}{3}(2{)}^{3/4}$

Q. Evaluate the limit:

$\underset{x\to 1}{lim}\{\frac{x-2}{x^2-x}-\frac{1}{x^3-3x^2+2x}\}$

Q. The value of $\underset{n\to \infty }{lim}\frac{1^1+2^2+3^3+\cdots +n^n}{1^n+2^n+3^n+\cdots +n^n}$ is

1. $\frac{e-1}{e}$
2. $\frac{e}{e-1}$
3. $\frac{1}{e-1}$
4. $\frac{e-1}{e+1}$

Q. $\underset{x\to 2}{lim}\left(\sum _{n=1}^9\frac{x}{n(n+1)x^2+2(2n+1)x+4}\right)$ is equal to:

1. $\frac{7}{36}$
2. $\frac{1}{5}$
3. $\frac{5}{24}$
4. $\frac{9}{44}$

Q. Evaluate the limit:
$\underset{x\to 0}{lim}\frac{8^x-4^x-2^x+1}{x^2}$

Q. The value of $\underset{n\to \infty }{lim}\frac{1\cdot 2+2\cdot 3+3\cdot 4+\cdots +n(n+1)}{n^3}$ is

1. $1$
2. $-1$
3. $\frac{1}{3}$
4. $-\frac{1}{3}$

Q.

Evaluate: ${lim}_{x\to 0}\frac{ax+b}{cx+d},d\ne 0$

Q. $\underset{n\to \infty }{lim}(\frac{(n+1{)}^{1/3}}{n^{4/3}}+\frac{(n+2{)}^{1/3}}{n^{4/3}}+\cdots +\frac{(2n{)}^{1/3}}{n^{4/3}})$ is equal to

1. $\frac{3}{4}(2{)}^{4/3}-\frac{3}{4}$
2. $\frac{4}{3}(2{)}^{4/3}$
3. $\frac{3}{4}(2{)}^{4/3}-\frac{4}{3}$
4. $\frac{4}{3}(2{)}^{3/4}$

Q.

If $U_n=\left|\begin{array}{ccc}n& 1& 5\\ n^2& 2N+1& 2N+1\\ n^3& 3N^2& 3N+1\end{array}\right|$ and $\sum _{n=1}^N{U}_n=\lambda \sum _{n=1}^N{n}^2,$ then the value of $\lambda$ is

Q. Evaluate the limit:
$\underset{x\to \infty }{lim}\frac{3x^{-1}+4x^{-2}}{5x^{-1}+6x^{-2}}$

Q. Let $\left[y\right]$ denote the greatest integer less than or equal to $y.$ If $f:(0,\infty )\to N$ is defined by $f\left(x\right)=\left[\frac{x^2+x+1}{x^2+1}\right]+\left[\frac{4x^2+x+2}{2x^2+1}\right]+\left[\frac{9x^2+x+3}{3x^2+1}\right]+\cdots +\left[\frac{n^2{x}^2+x+n}{n{x}^2+1}\right]$ for $n\in N,$ then the value of $\underset{n\to \infty }{lim}\left(\frac{f\left(x\right)-n}{{\left(f\right(x\left)\right)}^2-\frac{n^3(n+2)}{4}}\right)$ is

Q. The value that we must assign to $(cosx{)}^{\frac{1}{x}}$ at x = 0 so that the function will be continuous at x = 0 is:

1. -1
2. e
3. 0
4. 1

Q. $\begin{array}{c}lim\\ x\to a\end{array}\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}\left(from\frac{0}{0}\right)$

Q. If $I_n=\underset{0}{\overset{1}{\int }}\frac{dx}{(1+x^2{)}^n}$, where $n\in N$, then which of the following is/are correct ?

1. $2n{I}_{n+1}=2^{-n}+(2n-1)I_n$
2. $I_2=\frac{1}{4}+\frac{\pi }{8}$
3. $I_2=\frac{\pi }{8}-\frac{1}{4}$
4. $I_3=\frac{1}{4}+\frac{3\pi }{32}$

Q.

If ${△}_r=\left|\begin{array}{ccc}1& n& n\\ 2r& n^2+n+1& n^2+n\\ 2r-1& n^2& n^2+n+1\end{array}\right|$ and $\sum _{r=0}^m{△}_r=56$, then $n=$

1. $4$

2. $6$

3. $7$

4. $8$

Q. Evaluate $\underset{x\to \infty }{lim}\frac{5x^2+4}{\sqrt{2x^4+1}}$

Q. $\underset{n\to \infty }{lim}\frac{{}^n{P}_n}{{}^{n+1}{P}_n-{}^n{P}_n}=$

Q. If X is Poisson variate with P(X=0)=P(X=1), then P(X=2)=

1. $\frac{e}{2}$
2. $\frac{e}{6}$
3. $\frac{1}{2e}$
4. $\frac{1}{6e}$

Q. $\text{If}\underset{n\to \infty }{lim}\frac{(1^{\lambda }+2^{\lambda }+3^{\lambda }+--+n^{\lambda })}{(1^4+2^4+--+n^4)(1^3+2^3+--+n^3)}=F\left(\lambda \right),\lambda ϵN,\text{then}$

1. $F\left(\lambda \right)$ is finite for $\lambda \le 8$
2. $F\left(8\right)=\frac{20}{11}$
3. $F\left(8\right)=\frac{20}{9}$
4. $F\left(7\right)=0$

Q. Statement $1:$ Let $f\left(x\right)=\underset{m\to \infty }{lim}\left(\underset{n\to \infty }{lim}{\cos }^{2m}(n!\pi x)\right),$ and $g\left(x\right)=\{\begin{array}{c}0,\text{if}x\text{is rational}\\ 1,\text{if}x\text{is irrrational}\end{array}$. Then $h\left(x\right)=f\left(x\right)+g\left(x\right)$ is continuous for all $x.$
Statement $2:$ $f\left(x\right)$ and $g\left(x\right)$ are discontinuous functions for $x\in R.$

1. Both the statements are true and Statement $2$ is the correct explanation of Statement $1.$
2. Both the statements are true and Statement $2$ is not the correct explanation of Statement $1.$
3. Statement $1$ is true and Statement $2$ is false
4. Statement $1$ is false and Statement $2$ is true

Q. $\text{If}\underset{n\to \infty }{lim}\frac{(1^{\lambda }+2^{\lambda }+3^{\lambda }+--+n^{\lambda })}{(1^4+2^4+--+n^4)(1^3+2^3+--+n^3)}=F\left(\lambda \right),\lambda ϵN,\text{then}$

1. $F\left(\lambda \right)$ is finite for $\lambda \le 8$
2. $F\left(8\right)=\frac{20}{11}$
3. $F\left(8\right)=\frac{20}{9}$
4. $F\left(7\right)=0$

Q. Prove that :
$\underset{x\to 1}{lim}\frac{x^4-\sqrt{x}}{\sqrt{x}-1}=7$

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. $\underset{x\to \sqrt{2}}{lim}\frac{x^2-2}{x^2+3\sqrt{2}x-8}$