Limit $f(x)$ Evaluation
The function is defined as \( f(x) = \lim_{n \to \infty} \frac{1}{n^3} \sum_{k=1}^n \left[ \frac{k^2}{3^x} \right] \). We use the inequality \( y - 1 < [y] \le y \).
Applying this, we get bounds for the term inside the limit:
$ \frac{1}{n^3} \sum_{k=1}^n \left( \frac{k^2}{3^x} - 1 \right) < \frac{1}{n^3} \sum_{k=1}^n \left[ \frac{k^2}{3^x} \right] \le \frac{1}{n^3} \sum_{k=1}^n \frac{k^2}{3^x} $We evaluate the limits of the bounds as \( n \to \infty \). Using the formula \( \sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6} \), we find the limit:
$ \lim_{n \to \infty} \frac{1}{n^3} \sum_{k=1}^n \frac{k^2}{3^x} = \frac{1}{3^x} \lim_{n \to \infty} \frac{n(n+1)(2n+1)}{6n^3} $ $ = \frac{1}{3^x} \cdot \frac{1}{6} \lim_{n \to \infty} \frac{n^3(1+1/n)(2+1/n)}{n^3} = \frac{1}{3^x} \cdot \frac{1}{6} \cdot 2 = \frac{1}{3 \cdot 3^x} = \frac{1}{3^{x+1}} $The limit of the lower bound is also \( \frac{1}{3^{x+1}} \). By the Squeeze Theorem, \( f(x) = \frac{1}{3^{x+1}} \).
Summation $\sum f(j)$ Calculation
We need to compute the sum $S = \sum_{j=1}^\infty f(j)$:
$ S = \sum_{j=1}^\infty \frac{1}{3^{j+1}} $This is an infinite geometric series with the first term \( a = \frac{1}{3^{1+1}} = \frac{1}{9} \) and the common ratio \( r = \frac{1}{3} \).
The sum of an infinite geometric series is given by \( S = \frac{a}{1-r} \) (where \( |r| < 1 \)).
$ S = \frac{1/9}{1 - 1/3} = \frac{1/9}{2/3} = \frac{1}{9} \times \frac{3}{2} = \frac{1}{6} $Sum $12 \sum f(j)$
Finally, we calculate the required value:
$ 12 \times S = 12 \times \frac{1}{6} = 2 $The result is 2.
Similar Questions
- Let a differentiable function $f$ satisfy the equation $\int_{0}^{36} f\left(\frac{tx}{36}\right) dt = 4\alpha f(x)$. If $y = f(x)$ is a standard parabola passing through the points (2, 1) and (– 4, $\beta$), then $\beta^\alpha$ is equal to ______.
Let $f$ be a differentiable function satisfying $f(x) = 1 - 2x + \int_{0}^{x} e^{(x - t)} f(t) dt$, $x \in \mathbf{R}$ and let $g(x) = \int_{0}^{x} (f(t) + 2)^{15} (t - 4)^{6} (t + 12)^{17} dt$, $x \in \mathbf{R}$. If $\text{p}$ and $\text{q}$ are respectively the points of local minima and local maxima of $g$, then the value of $|\text{p} + \text{q}|$ is equal to _________.
Let $f$ be a twice differentiable function such that $f(x) = \int_{0}^{x} \tan(t - x)dt - \int_{0}^{x} f(t)\tan t dt$, $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.
Then $f''\left(\frac{\pi}{6}\right) + 12f'\left(-\frac{\pi}{6}\right) + f\left(\frac{\pi}{6}\right)$ is equal to __________- Let $y = y(x)$ be the solution of the differential equation $\left(x^2 - x\sqrt{x^2 - 1}\right)dy + \left(y\left(x - \sqrt{x^2 - 1}\right) - x\right)dx = 0, x \geq 1$. If $y(1) = 1$, then the greatest integer less than $y\left(\sqrt{5}\right)$ is _________.
- If $I_1 = \int_0^1 e^{-x} cos^2x dx$, $I_2 = \int_0^1 e^{-x^2} cos^2x dx$ and $I_3 = \int_0^1 e^{-x^2} dx$; then :
- If $\int \frac{(\sqrt{1+x^2}+x)^{10}}{(\sqrt{1+x^2}-x)^9} dx = \frac{1}{m} \left( (\sqrt{1+x^2}+x)^n (n\sqrt{1+x^2}-x) \right) + C$ where $C$ is the constant of integration and $m, n \in N$, then $m+n$ is equal to
Let [.] denote the greatest integer function. If $\int_{0}^{e^3} \left[\frac{1}{e^{x-1}}\right] dx = \alpha - \log_e 2$, then $\alpha^3$ is equal to ____________.
Let $f: R\to R$ be a thrice differentiable odd function satisfying $f'(x)\ge0, f''(x)=f(x), f(0)=0, f'(0)=3$. Then $9f(\log_e 3)$ is equal to ___________.
- Let $y = y (x)$ be the solution curve of the differentialequation $x (x^2 + e^x) dy + (e^x (x-2) y-x^3) dx = 0, x > 0$, passing through the point $(1, 0)$.Then $y (2)$ is equal to
$4\int_{0}^{1} (\frac{1}{\sqrt{3+x^2} + \sqrt{1+x^2}}) dx - 3\log_e (\sqrt{3})$ is equal to :
Important Questions from Integral Calculus
- Let a differentiable function $f$ satisfy the equation $\int_{0}^{36} f\left(\frac{tx}{36}\right) dt = 4\alpha f(x)$. If $y = f(x)$ is a standard parabola passing through the points (2, 1) and (– 4, $\beta$), then $\beta^\alpha$ is equal to ______.
Let $f$ be a differentiable function satisfying $f(x) = 1 - 2x + \int_{0}^{x} e^{(x - t)} f(t) dt$, $x \in \mathbf{R}$ and let $g(x) = \int_{0}^{x} (f(t) + 2)^{15} (t - 4)^{6} (t + 12)^{17} dt$, $x \in \mathbf{R}$. If $\text{p}$ and $\text{q}$ are respectively the points of local minima and local maxima of $g$, then the value of $|\text{p} + \text{q}|$ is equal to _________.
Let $f$ be a twice differentiable function such that $f(x) = \int_{0}^{x} \tan(t - x)dt - \int_{0}^{x} f(t)\tan t dt$, $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.
Then $f''\left(\frac{\pi}{6}\right) + 12f'\left(-\frac{\pi}{6}\right) + f\left(\frac{\pi}{6}\right)$ is equal to __________- Let $y = y(x)$ be the solution of the differential equation $\left(x^2 - x\sqrt{x^2 - 1}\right)dy + \left(y\left(x - \sqrt{x^2 - 1}\right) - x\right)dx = 0, x \geq 1$. If $y(1) = 1$, then the greatest integer less than $y\left(\sqrt{5}\right)$ is _________.
- If $I_1 = \int_0^1 e^{-x} cos^2x dx$, $I_2 = \int_0^1 e^{-x^2} cos^2x dx$ and $I_3 = \int_0^1 e^{-x^2} dx$; then :
