Let [·] denote the greatest integer function and f(x) = lim(n→∞) (1/n³)Σ[k²/3ˣ]. Then 12Σf(j) from j=1 to ∞ is equal to

The greatest integer function, denoted by $[x]$ or $\lfloor x \rfloor$, returns the largest integer less than or equal to $x$. For instance, $[4.7] = 4$, $[5] = 5$, $[-2.3] = -3$, and $[0.99] = 0$. This function is discontinuous at every integer value and has a step-like graph. In calculus problems, when dealing with limits involving the greatest integer function, we often approximate $[f(x)] \approx f(x)$ when $f(x)$ is much larger than 1, or when we’re summing over many terms where the discretization error becomes negligible. The greatest integer function satisfies the property that $x – 1 < [x] \leq x$ for all real $x$. In JEE problems, it frequently appears in combination with limits, integration, and series, requiring careful analysis of whether the integer part affects the limiting behavior. For large arguments, especially in Riemann sums, the contribution of the fractional part becomes negligible, allowing us to replace $[f(x)]$ with $f(x)$ in the limit. This approximation is valid because when summing over $n$ terms with each having an error of at most 1, the total error is bounded by $n$, but when divided by $n^3$ (as in our problem), this error vanishes as $n \to \infty$.

A Riemann sum is a method of approximating the total area underneath a curve on a graph, which is the definite integral of the function. The basic idea is to divide the region into rectangles and sum their areas. For a function $f(x)$ on the interval $[a,b]$, we partition the interval into $n$ subintervals of equal width $\Delta x = \frac{b-a}{n}$. The Riemann sum is $S_n = \sum_{i=1}^{n} f(x_i^*) \Delta x$, where $x_i^*$ is a sample point in the $i$-th subinterval. As $n \to \infty$, this sum converges to the definite integral: $\lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x = \int_a^b f(x) dx$. In the standard form for JEE, we often see $\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{n} f\left(\frac{r}{n}\right) = \int_0^1 f(x) dx$. This is because $\Delta x = \frac{1}{n}$ and $x_i = \frac{i}{n}$. The substitution $t = \frac{k}{n}$ transforms discrete sums into continuous integrals. When the summand involves $\frac{1}{n^p}$ for $p > 1$, we can factor it appropriately. For instance, $\frac{1}{n^3} \sum_{k=1}^{n} g(k) = \frac{1}{n} \sum_{k=1}^{n} \frac{g(k)}{n^2}$, which with $k = nt$ gives us $\int_0^1 \frac{g(nt)}{n^2} dt$. Understanding this transformation is crucial for converting complex summations into tractable integrals that can be evaluated using standard calculus techniques.

A geometric series is a series of the form $\sum_{n=0}^{\infty} ar^n = a + ar + ar^2 + ar^3 + \ldots$, where $a$ is the first term and $r$ is the common ratio. The series converges if and only if $|r| < 1$, and when it converges, its sum is given by the formula $S = \frac{a}{1-r}$. If we start from $n=1$ instead of $n=0$, we have $\sum_{n=1}^{\infty} ar^n = ar + ar^2 + \ldots = \frac{ar}{1-r}$. In our problem, the series $\sum_{j=1}^{\infty} \frac{1}{3^j}$ is geometric with $a = \frac{1}{3}$ and $r = \frac{1}{3}$, giving sum $\frac{1/3}{1-1/3} = \frac{1}{2}$. Geometric series are fundamental in analyzing sequences and series in JEE Mathematics. They appear in various contexts including probability (geometric distribution), financial mathematics (present value calculations), and physics (infinite reflections, radioactive decay). The convergence criterion $|r| < 1$ is essential—if $|r| \geq 1$, the series diverges. For $|r| < 1$, the terms decrease exponentially, ensuring convergence. The partial sum formula $S_n = a\frac{1-r^n}{1-r}$ shows that as $n \to \infty$ and $|r| < 1$, we have $r^n \to 0$, giving the infinite sum formula. Understanding geometric series also helps in solving recurrence relations and analyzing algorithms in computer science applications.

Integration of polynomial functions is one of the most fundamental operations in calculus. The power rule for integration states that $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$. For definite integrals over an interval $[a,b]$, we use the fundamental theorem of calculus: $\int_a^b f(x)dx = F(b) – F(a)$, where $F$ is an antiderivative of $f$. In our problem, we evaluated $\int_0^1 t^2 dt = \frac{t^3}{3}\Big|_0^1 = \frac{1}{3} – 0 = \frac{1}{3}$. Common polynomial integrals include: $\int_0^1 x dx = \frac{1}{2}$, $\int_0^1 x^2 dx = \frac{1}{3}$, $\int_0^1 x^3 dx = \frac{1}{4}$, and in general $\int_0^1 x^n dx = \frac{1}{n+1}$. These results are frequently used in physics for calculating centers of mass, moments of inertia, and work done by variable forces. When integrating polynomials, we can use linearity: $\int [af(x) + bg(x)]dx = a\int f(x)dx + b\int g(x)dx$. This allows us to break complex polynomials into simpler terms. For example, $\int (3x^2 + 2x – 1)dx = 3\int x^2 dx + 2\int x dx – \int 1 dx = x^3 + x^2 – x + C$. The constant of integration $C$ is crucial for indefinite integrals but disappears in definite integrals due to subtraction. Mastering polynomial integration is essential as it forms the foundation for more advanced techniques like substitution, integration by parts, and partial fractions.

Limits are fundamental to calculus, providing the rigorous foundation for derivatives and integrals. The limit $\lim_{x \to a} f(x) = L$ means that $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $a$. Key limit properties include: linearity ($\lim[af(x) + bg(x)] = a\lim f(x) + b\lim g(x)$), product rule ($\lim[f(x)g(x)] = \lim f(x) \cdot \lim g(x)$), and quotient rule ($\lim\frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)}$ if $\lim g(x) \neq 0$). For limits at infinity, we often factor out the highest power term. For instance, $\lim_{n \to \infty} \frac{3n^2 + 2n}{5n^2 – 1} = \lim_{n \to \infty} \frac{3 + 2/n}{5 – 1/n^2} = \frac{3}{5}$. Important standard limits include: $\lim_{x \to 0} \frac{\sin x}{x} = 1$, $\lim_{x \to 0} \frac{e^x – 1}{x} = 1$, $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$, and $\lim_{n \to \infty} \frac{1}{n^p} = 0$ for $p > 0$. The squeeze theorem is powerful: if $g(x) \leq f(x) \leq h(x)$ and $\lim g(x) = \lim h(x) = L$, then $\lim f(x) = L$. L’Hôpital’s rule handles indeterminate forms like $\frac{0}{0}$ and $\frac{\infty}{\infty}$: if $\lim \frac{f(x)}{g(x)}$ gives such a form, then $\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}$ (if the latter exists). In series problems, limits help determine convergence. The connection between limits and Riemann sums transforms discrete problems into continuous ones, making complex summations tractable through integration.

Riemann Sum to Integral Geometric Series Formula Power Rule Integration Greatest Integer Properties Limit at Infinity Series Convergence Tests