Integral as the limit of a sum — practice questions

Question 1 · difficulty L3 · understanding

Let $[\cdot]$ denote the greatest integer function and $f(x) = \lim\limits_{n \to \infty} \frac{1}{n^{3}} \sum\limits_{k=1}^n \left[ \frac{k^2}{3^x} \right]$. Then $12 \sum\limits_{j=1}^{\infty} f(i)$ is equal to ________.

Question 2 · difficulty L3 · understanding

$$\begin{aligned} &\text { If } \lim _{n \rightarrow \infty} \frac{(n+1)^{k-1}}{n^{k+1}}[(n k+1)+(n k+2)+\ldots+(n k+n)] \\ &=33 \cdot \lim _{n \rightarrow \infty} \frac{1}{n^{k+1}} \cdot\left[1^{k}+2^{k}+3^{k}+\ldots+n^{k}\right] \end{aligned}$$, then the integral value of $$\mathrm{k}$$ is equal to _____________

Question 3 · difficulty L3 · understanding

Let f : (0, 2) $$\to$$ R be defined as f(x) = log 2 $$\left( {1 + \tan \left( {{{\pi x} \over 4}} \right)} \right)$$. Then, $$\mathop {\lim }\limits_{n \to \infty } {2 \over n}\left( {f\left( {{1 \over n}} \right) + f\left( {{2 \over n}} \right) + ... + f(1)} \right)$$ is equal to ___________.

Question 4 · difficulty L3 · understanding

The value of \lim _\limits{n \rightarrow \infty} \sum_\limits{k=1}^n \frac{n^3}{\left(n^2+k^2\right)\left(n^2+3 k^2\right)} is :

• A. $$\frac{\pi}{8(2 \sqrt{3}+3)}$$
• B. $$\frac{(2 \sqrt{3}+3) \pi}{24}$$
• C. $$\frac{13 \pi}{8(4 \sqrt{3}+3)}$$
• D. $$\frac{13(2 \sqrt{3}-3) \pi}{8}$$

Question 5 · difficulty L3 · understanding

Among (S1): \lim_\limits{n \rightarrow \infty} \frac{1}{n^{2}}(2+4+6+\ldots \ldots+2 n)=1 (S2) : \lim_\limits{n \rightarrow \infty} \frac{1}{n^{16}}\left(1^{15}+2^{15}+3^{15}+\ldots \ldots+n^{15}\right)=\frac{1}{16}

• A. Only (S1) is true
• B. Both (S1) and (S2) are true
• C. Both (S1) and (S2) are false
• D. Only (S2) is true

Question 6 · difficulty L3 · understanding

$\lim\limits_{n \rightarrow \infty} \frac{3}{n}\left\{4+\left(2+\frac{1}{n}\right)^2+\left(2+\frac{2}{n}\right)^2+\ldots+\left(3-\frac{1}{n}\right)^2\right\}$ is equal to :

• A. 0
• B. $\frac{19}{3}$
• C. 19
• D. 12

Question 7 · difficulty L3 · understanding

If $$a = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {{{2n} \over {{n^2} + {k^2}}}}$$ and $$f(x) = \sqrt {{{1 - \cos x} \over {1 + \cos x}}}$$, $$x \in (0,1)$$, then :

• A. $$2\sqrt 2 f\left( {{a \over 2}} \right) = f'\left( {{a \over 2}} \right)$$
• B. $$f\left( {{a \over 2}} \right)f'\left( {{a \over 2}} \right) = \sqrt 2$$
• C. $$\sqrt 2 f\left( {{a \over 2}} \right) = f'\left( {{a \over 2}} \right)$$
• D. $$f\left( {{a \over 2}} \right) = \sqrt 2 f'\left( {{a \over 2}} \right)$$

Question 8 · difficulty L3 · understanding

$$\mathop {\lim }\limits_{n \to \infty } {1 \over {{2^n}}}\left( {{1 \over {\sqrt {1 - {1 \over {{2^n}}}} }} + {1 \over {\sqrt {1 - {2 \over {{2^n}}}} }} + {1 \over {\sqrt {1 - {3 \over {{2^n}}}} }} + \,\,...\,\, + \,\,{1 \over {\sqrt {1 - {{{2^n} - 1} \over {{2^n}}}} }}} \right)$$ is equal to

• A. $$\frac{1}{2}$$
• B. 1
• C. 2
• D. $$-$$2

Question 9 · difficulty L3 · understanding

$$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{r \over {2{r^2} - 7rn + 6{n^2}}}}$$ is equal to :

• A. $${\log _e}\left( {{{\sqrt 3 } \over 2}} \right)$$
• B. $${\log _e}\left( {{{3\sqrt 3 } \over 4}} \right)$$
• C. $${\log _e}\left( {{{27} \over 4}} \right)$$
• D. $${\log _e}\left( {{4 \over 3}} \right)$$

Question 10 · difficulty L3 · understanding

If $${U_n} = \left( {1 + {1 \over {{n^2}}}} \right)\left( {1 + {{{2^2}} \over {{n^2}}}} \right)^2.....\left( {1 + {{{n^2}} \over {{n^2}}}} \right)^n$$, then $$\mathop {\lim }\limits_{n \to \infty } {({U_n})^{{{ - 4} \over {{n^2}}}}}$$ is equal to :

• A. $${{{e^2}} \over {16}}$$
• B. $${4 \over e}$$
• C. $${{16} \over {{e^2}}}$$
• D. $${4 \over {{e^2}}}$$