GATE Statistics Calculus Syllabus

State the Ratio Test (D'Alembert) for $\sum a_n$.

Let $L = \lim_{n\to\infty} \left| \dfrac{a_{n+1}}{a_n} \right|$. If $L < 1$ the series converges absolutely; if $L > 1$ it diverges; if $L = 1$ the test is inconclusive.

State the Root Test (Cauchy) for $\sum a_n$.

Let $L = \limsup_{n\to\infty} |a_n|^{1/n}$. If $L < 1$ the series converges absolutely; if $L > 1$ it diverges; if $L = 1$ the test is inconclusive.

State the Integral Test for series.

If $f$ is positive, continuous and decreasing on $[1,\infty)$ with $f(n) = a_n$, then $\sum_{n=1}^{\infty} a_n$ and $\int_{1}^{\infty} f(x)\,dx$ either both converge or both diverge.

State Cauchy's condensation test.

For a non-increasing sequence $a_n \geq 0$, $\sum_{n=1}^{\infty} a_n$ converges iff $\sum_{k=0}^{\infty} 2^{k} a_{2^{k}}$ converges.

State Raabe's test for $\sum a_n$ (positive terms).

Let $L = \lim_{n\to\infty} n\left( \dfrac{a_n}{a_{n+1}} - 1 \right)$. If $L > 1$ the series converges; if $L < 1$ it diverges; $L = 1$ is inconclusive. Useful when the ratio test gives $1$.

State the Alternating Series (Leibniz) Test.

The series $\sum_{n=1}^{\infty} (-1)^{n+1} b_n$ with $b_n > 0$ converges if $b_n$ is monotonically decreasing and $b_n \to 0$.

State the error bound for an alternating series satisfying Leibniz's conditions.

If $S$ is the sum and $S_N$ the $N$-th partial sum, then $|S - S_N| \leq b_{N+1}$: the truncation error is bounded by the first omitted term.

Define absolute convergence of a series.

$\sum a_n$ converges absolutely if $\sum |a_n|$ converges. Absolute convergence implies convergence of $\sum a_n$.

Define conditional convergence and give a standard example.

$\sum a_n$ converges conditionally if $\sum a_n$ converges but $\sum |a_n|$ diverges. Example: $\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}}{n}$ converges (to $\ln 2$) but $\sum \frac{1}{n}$ diverges.

What does Riemann's rearrangement theorem say about conditionally convergent series?

A conditionally convergent series can be rearranged to converge to any real number, or to diverge. Absolutely convergent series have the same sum under any rearrangement.

Give the $\varepsilon$-$\delta$ definition of $\lim_{x \to a} f(x) = L$.

For every $\varepsilon > 0$ there exists $\delta > 0$ such that $0 < |x - a| < \delta$ implies $|f(x) - L| < \varepsilon$.

State the relationship between one-sided limits and the existence of a limit.

$\lim_{x \to a} f(x)$ exists and equals $L$ iff both one-sided limits exist and are equal: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$.

Give the $\varepsilon$-$\delta$ definition of continuity of $f$ at $a$.

$f$ is continuous at $a$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that $|x - a| < \delta$ implies $|f(x) - f(a)| < \varepsilon$; equivalently $\lim_{x\to a} f(x) = f(a)$.

State the Intermediate Value Theorem.

If $f$ is continuous on $[a,b]$ and $k$ lies between $f(a)$ and $f(b)$, then there exists $c \in (a,b)$ with $f(c) = k$.

State the Extreme Value Theorem (boundedness theorem).

A function continuous on a closed bounded interval $[a,b]$ is bounded and attains its maximum and minimum values on $[a,b]$.

Define a monotone (monotonic) function on an interval.

$f$ is increasing if $x_1 < x_2 \Rightarrow f(x_1) \leq f(x_2)$ and decreasing if $x_1 < x_2 \Rightarrow f(x_1) \geq f(x_2)$. Strict versions use strict inequalities on the values.

What kind of discontinuities can a monotone function on an interval have?

A monotone function can have at most countably many discontinuities, and they are all jump (first-kind) discontinuities; it has no removable or essential discontinuities.

Define uniform continuity of $f$ on a set $A$.

$f$ is uniformly continuous on $A$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that for all $x, y \in A$, $|x - y| < \delta \Rightarrow |f(x) - f(y)| < \varepsilon$ ($\delta$ independent of the point).

State the theorem linking continuity on a closed interval to uniform continuity (Heine–Cantor).

A function continuous on a closed and bounded interval $[a,b]$ is uniformly continuous on $[a,b]$.

Give an example showing continuity does not imply uniform continuity, and explain.

$f(x) = \frac{1}{x}$ on $(0,1)$ and $f(x) = x^{2}$ on $\mathbb{R}$ are continuous but not uniformly continuous, because the required $\delta$ shrinks to $0$ as $x$ approaches $0$ (resp. grows large).

Define differentiability of $f$ at $a$.

$f$ is differentiable at $a$ if $f'(a) = \lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$ exists (finitely).

What is the relationship between differentiability and continuity?

Differentiability at $a$ implies continuity at $a$, but not conversely. Example: $f(x) = |x|$ is continuous at $0$ but not differentiable there.

State Rolle's Theorem with all hypotheses.

If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then there exists $c \in (a,b)$ such that $f'(c) = 0$.

State Lagrange's Mean Value Theorem.

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c \in (a,b)$ with $f'(c) = \dfrac{f(b) - f(a)}{b - a}$.

State Cauchy's Mean Value Theorem (generalized MVT).

If $f, g$ are continuous on $[a,b]$, differentiable on $(a,b)$, and $g'(x) \neq 0$, then there exists $c \in (a,b)$ with $\dfrac{f'(c)}{g'(c)} = \dfrac{f(b) - f(a)}{g(b) - g(a)}$.

What does the sign of $f'$ tell you about monotonicity (a corollary of the MVT)?

If $f'(x) > 0$ on an interval then $f$ is strictly increasing there; if $f'(x) < 0$ then strictly decreasing; if $f'(x) = 0$ throughout then $f$ is constant.

State Taylor's Theorem with the Lagrange form of the remainder.

If $f$ is $n+1$ times differentiable, then $f(x) = \sum_{k=0}^{n} \dfrac{f^{(k)}(a)}{k!}(x-a)^{k} + R_n$, where $R_n = \dfrac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}$ for some $\xi$ between $a$ and $x$.

Write the Maclaurin series (Taylor at $0$) for $e^{x}$.

$$e^{x} = \sum_{n=0}^{\infty} \frac{x^{n}}{n!} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots$$

State L'Hôpital's Rule for the $\frac{0}{0}$ indeterminate form.

If $\lim_{x\to a} f(x) = \lim_{x\to a} g(x) = 0$, $f,g$ differentiable near $a$ with $g'(x) \neq 0$, and $\lim_{x\to a}\frac{f'(x)}{g'(x)}$ exists, then $\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$.

To which indeterminate forms does L'Hôpital's Rule directly apply, and how are others handled?

It applies directly to $\frac{0}{0}$ and $\frac{\infty}{\infty}$. Forms $0\cdot\infty$ and $\infty-\infty$ are converted to one of these by algebra; $0^{0}$, $1^{\infty}$, $\infty^{0}$ are handled by taking logarithms first.