Property:

Limit as a sum

• Express the given series in the form of $\sum {\displaystyle \frac{1}{\mathrm{n}}}f\left({\displaystyle \frac{\mathrm{r}}{\mathrm{n}}}\right)$

• Then the limit as its sum when $\mathrm{n}\to \mathrm{\infty }$ i.e. $\underset{\mathrm{n}\to \mathrm{\infty }}{lim}\sum {\displaystyle \frac{1}{\mathrm{n}}}.f\left({\displaystyle \frac{\mathrm{r}}{\mathrm{n}}}\right)$

• Replace ${\displaystyle \frac{\mathrm{r}}{\mathrm{n}}}$ by $x$ and ${\displaystyle \frac{1}{\mathrm{n}}}$ by $\mathrm{dx}$ and ${\displaystyle \frac{1}{\mathrm{n}}}$ by the sign of $\int$

Solved Examples

1. $\underset{\mathrm{n}\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{\mathrm{n}+1}}+{\displaystyle \frac{1}{\mathrm{n}+2}}+{\displaystyle \frac{1}{\mathrm{n}+3}}+—-+{\displaystyle \frac{1}{2\mathrm{n}}}=$

(a) 1

(b) 2

(c) ${\log }_{\mathrm{e}}2$

(d) ${\log }_{e}4$

2. The value of $\sum _{\mathrm{r}=0}^{\mathrm{n}-1}{\displaystyle \frac{1}{\sqrt{4{\mathrm{n}}^2-{\mathrm{r}}^2}}}$ as $\mathrm{n}\to \mathrm{\infty }$ is

(a) ${\displaystyle \frac{\pi }{3}}$

(b) ${\displaystyle \frac{\pi }{6}}$

(c) ${\displaystyle \frac{\pi }{4}}$

(d) none of these

3. The value of $\underset{\mathrm{n}\to \mathrm{\infty }}{lim}{\displaystyle \frac{(1^2+2^2+3^2+-+{\mathrm{n}}^2)(1^3+2^3+-+{\mathrm{n}}^3)}{1^6+2^6+—–+{\mathrm{n}}^6}}$

(a) 1

(b) ${\displaystyle \frac{7}{12}}$

(c) ${\displaystyle \frac{5}{6}}$

(d) none

4. ${\int }_0^1\left[{\displaystyle \frac{2}{{\mathrm{e}}^{\mathrm{x}}}}\right]\mathrm{dx}$ is equal to

(a) ${\log }_{e}2$

(b) ${\mathrm{e}}^2$

(c) 0

(d) ${\displaystyle \frac{2}{\mathrm{e}}}$

5. The value of ${\int }_0^{\pi /3}[\sqrt{3}\tan x]dx$ is

(a) ${\displaystyle \frac{5\pi }{6}}$

(b) ${\displaystyle \frac{5\pi }{6}}-{\tan }^{-1}{\displaystyle \frac{2}{\sqrt{3}}}$

(c) ${\displaystyle \frac{\pi }{2}}-{\tan }^{-1}{\displaystyle \frac{2}{\sqrt{3}}}$

(d) none

6. ${\int }_{-1}^1[\mathrm{x}[1+\sin \pi \mathrm{x}]+1]\mathrm{dx}$ is equal to

(a) 2

(b) 0

(c) 1

(d) none

7. ${\int }_0^2[{\mathrm{x}}^2-1]\mathrm{dx}$ is equal to

(a) $3-\sqrt{3}-\sqrt{2}$

(b) 2

(c) 1

(d) $3-2\sqrt{3}-3\sqrt{2}$

Exercise

1. $\underset{\mathrm{n}\to \mathrm{\infty }}{lim}{\displaystyle \frac{1^{\mathrm{P}}+2^{\mathrm{P}}+3^{\mathrm{P}}+\dots \dots +{\mathrm{n}}^{\mathrm{P}}}{{\mathrm{n}}^{\mathrm{P}}+1}}=$

(a) ${\displaystyle \frac{1}{\mathrm{P}+1}}$

(b) ${\displaystyle \frac{1}{1-\mathrm{P}}}$

(c) ${\displaystyle \frac{1}{\mathrm{P}}}-{\displaystyle \frac{1}{\mathrm{P}-1}}$

(d) ${\displaystyle \frac{\mathrm{P}}{\mathrm{P}+1}}$

2. Let a,b,c be non-zero real numbers such that ${\int }_0^1(1+{\cos }^8\mathrm{x})(a{x}^2+\mathrm{bx}+\mathrm{c})\mathrm{dx}$

$={\int }_0^2(1+{\cos }^8\mathrm{x})({\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c})\mathrm{dx}$. Then the quadratic equation ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$ has

(a) no root in $(0,2)$

(b) at least one root in $(1,2)$

(c) a double root in $(0,2)$

(d) two imaginary roots

3. For every function $f(\mathrm{x})$ which is twice differentiable these will be good approximation of

$\begin{array}{rl}& {\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{x})\mathrm{dx}={\displaystyle \frac{\mathrm{b}-\mathrm{a}}{2}}(f(\mathrm{a})+f(\mathrm{b})),\text{ for more accurate results for }\mathrm{c}\in (\mathrm{a},\mathrm{b})\\ & \mathrm{F}(\mathrm{c})={\displaystyle \frac{\mathrm{c}-\mathrm{a}}{2}}(f(\mathrm{a})-f(\mathrm{c}))+{\displaystyle \frac{\mathrm{b}-\mathrm{c}}{2}}(f(\mathrm{b})-f(\mathrm{c}))\text{ where }\mathrm{c}={\displaystyle \frac{\mathrm{a}+\mathrm{b}}{2}}\\ & {\int }_{\mathrm{a}}^{\mathrm{b}}f(\mathrm{x})\mathrm{dx}={\displaystyle \frac{\mathrm{b}-\mathrm{a}}{4}}(f(\mathrm{a})+f(\mathrm{b})+2f(\mathrm{c}))\mathrm{dx}\end{array}$

i. Good approximation of ${\int }_0^{\pi /2}\sin xdx$ is

(a) ${\displaystyle \frac{\pi }{4}}$

(b) ${\displaystyle \frac{\pi (\sqrt{2}+1)}{4}}$

(c) ${\displaystyle \frac{\pi (\sqrt{2}+1)}{8}}$

(d) ${\displaystyle \frac{\pi }{8}}$

ii. If $f^{\prime \prime }(\mathrm{x})<0,\mathrm{\forall }\mathrm{x}\in (\mathrm{a},\mathrm{b})$ & $(\mathrm{c},f(\mathrm{c}))$ is point of maximum where $\mathrm{c}\in (\mathrm{a}.\mathrm{b})$ then $f^{\prime }(\mathrm{c})$ is

(a) ${\displaystyle \frac{f(\mathrm{b})-f(\mathrm{a})}{\mathrm{b}-\mathrm{a}}}$

(b) $3\left({\displaystyle \frac{f(\mathrm{b})-f(\mathrm{a})}{\mathrm{b}-\mathrm{a}}}\right)$

(c) $2\left({\displaystyle \frac{f(\mathrm{b})-f(\mathrm{a})}{\mathrm{b}-\mathrm{a}}}\right)$

(d) 0

iii. If $\underset{\mathrm{t}\to \mathrm{a}}{lim}{\displaystyle \frac{\underset{\mathrm{a}}{\overset{\mathrm{t}}{\int }}f(\mathrm{x})\mathrm{dx}-{\displaystyle \frac{\mathrm{t}-\mathrm{a}}{2}}(f(\mathrm{t})+f(\mathrm{a}))}{(\mathrm{t}-\mathrm{a}{)}^3}}=0$, then degree of polynomial function $f(\mathrm{x})$ at most is

(a) 0

(b) 1

(c) 3

(d) 2

4. Match the following:-

5. ${\int }_1^3{\displaystyle \frac{dx}{x^2+[x{]}^2+1-2x[x]}}=$

(a) $\pi$

(b) $3\pi$

(c) $8\pi$

(d) ${\displaystyle \frac{\pi }{2}}$

6.* If $I_n={\int }_{-\pi }^{\pi }{\displaystyle \frac{\sin nx}{(1+{\pi }^x)\sin x}}dx,n=0,1,2,\dots \dots$ then

(a) $I_n=I_{n+2}$

(b) $\sum _{\mathrm{m}=1}^{10}{\mathrm{I}}_{2\mathrm{m}+1}=10\pi$

(c) $\sum _{\mathrm{m}=1}^{10}{\mathrm{I}}_{2\mathrm{m}}=0$

(d) $I_n=I_{n+1}$

7. All the values of a for which ${\int }_1^2(a^2+(4-4a)x+4x^3)dx\le 12$ are given by

(a) $\mathrm{a}\le 4$

(b) $a=3$

(c) $0\le \mathrm{a}<3$

(d) none of these

8. Value of ${\int }_0^2[x^2-x+1]dx$ is

(a) ${\displaystyle \frac{7+\sqrt{5}}{2}}$

(b) ${\displaystyle \frac{5-\sqrt{5}}{2}}$

(c) ${\displaystyle \frac{7-\sqrt{5}}{2}}$

(d) none of these

9. If $I={\int }_0^1{\displaystyle \frac{dx}{(5+2x-2x^2)(1+e^{2-4x})}}$ and $I_1={\int }_0^1{\displaystyle \frac{dx}{5+2x-2x^2}}dx$ then

(a) $\mathrm{I}={\displaystyle \frac{1}{2}}{\mathrm{I}}_1$

(b) $\mathrm{I}=2{\mathrm{I}}_1$

(c) $3\mathrm{I}=5{\mathrm{I}}_1$

(d) none of these

10. Match the following:-

11. ${\int }_0^2[{\tan }^{-1}x]dx+{\int }_0^2[{\cot }^{-1}x]dx=$

(a) $1-\cot 2$

(b) $1+\cot 2$

(c) $2(1+\cot 2)$

(d) $2(1-\cot 2)$

12.* Match the following

([.] denotes the greatest integer function)