CSIR NET Mathematical Science UNIT – 1 Syllabus

State the Ratio Test for the series $\sum a_n$.

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

State the Root Test for the series $\sum a_n$.

Let $L = \limsup_{n\to\infty}\sqrt[n]{|a_n|}$. If $L < 1$ converges absolutely; if $L > 1$ diverges; if $L = 1$ inconclusive.

For which values of $p$ does the $p$-series $\sum_{n=1}^{\infty} \frac{1}{n^{p}}$ converge?

It converges if and only if $p > 1$, and diverges for $p \leq 1$.

State the Leibniz (alternating series) test.

If $b_n \geq 0$ is decreasing with $b_n \to 0$, then the alternating series $\sum (-1)^{n} b_n$ converges. Also the remainder is bounded by the first omitted term.

Distinguish absolute convergence from conditional convergence of a series.

$\sum a_n$ converges absolutely if $\sum |a_n|$ converges. It converges conditionally if $\sum a_n$ converges but $\sum |a_n|$ diverges (e.g. $\sum (-1)^n/n$). Absolute convergence implies convergence.

Give the formula for the sum of a convergent geometric series and its convergence condition.

For $|r| < 1$, $\sum_{n=0}^{\infty} a r^{n} = \frac{a}{1-r}$. It diverges for $|r| \geq 1$.

State the epsilon-delta definition of continuity of $f$ at a point $c$.

$f$ is continuous at $c$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that $|x - c| < \delta \implies |f(x) - f(c)| < \varepsilon$.

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.

If $f$ is continuous on a closed bounded interval $[a,b]$, then $f$ attains a maximum and a minimum value on $[a,b]$.

Define uniform continuity of $f$ on a set $S$ and contrast it with ordinary continuity.

$f$ is uniformly continuous on $S$ if for every $\varepsilon > 0$ there is a single $\delta > 0$ such that for all $x,y \in S$, $|x-y| < \delta \implies |f(x)-f(y)| < \varepsilon$. The $\delta$ does not depend on the point, unlike ordinary continuity.

State the theorem linking continuity on a compact set to uniform continuity.

If $f$ is continuous on a compact set (e.g. a closed bounded interval $[a,b]$), then $f$ is uniformly continuous there (Heine-Cantor theorem).

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

$f(x) = \frac{1}{x}$ on $(0,1)$ is continuous but not uniformly continuous; the required $\delta$ shrinks to $0$ as $x \to 0^{+}$. Also $f(x)=x^2$ on $\mathbb{R}$ is continuous but not uniformly continuous.

Define differentiability of $f$ at a point $c$.

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

State the relationship between differentiability and continuity.

If $f$ is differentiable at $c$, then $f$ is continuous at $c$. The converse is false (e.g. $f(x)=|x|$ at $x=0$).

State Rolle's theorem.

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

State the (Lagrange) Mean Value Theorem.

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

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

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

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

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

How does the sign of $f'$ classify monotonicity of $f$ on an interval?

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

Define pointwise convergence of a sequence of functions $(f_n)$ to $f$ on a set $E$.

$f_n \to f$ pointwise on $E$ if for each $x \in E$ and each $\varepsilon > 0$ there is $N$ (depending on $x$) with $|f_n(x) - f(x)| < \varepsilon$ for all $n \geq N$.

Define uniform convergence of $(f_n)$ to $f$ on a set $E$.

$f_n \to f$ uniformly on $E$ if for every $\varepsilon > 0$ there is $N$ (independent of $x$) with $|f_n(x) - f(x)| < \varepsilon$ for all $n \geq N$ and all $x \in E$. Equivalently $\sup_{x\in E}|f_n(x)-f(x)| \to 0$.

State the theorem on uniform convergence preserving continuity.

If each $f_n$ is continuous on $E$ and $f_n \to f$ uniformly on $E$, then the limit $f$ is continuous on $E$.

State the theorem on interchanging limit and integral under uniform convergence.

If $f_n \to f$ uniformly on $[a,b]$ and each $f_n$ is integrable, then $\int_a^b f_n \to \int_a^b f$, i.e. $\lim_n \int_a^b f_n = \int_a^b \lim_n f_n$.

State the theorem for term-by-term differentiation of $(f_n)$.

If $f_n \to f$ pointwise on $[a,b]$, each $f_n$ is differentiable, and $f_n'$ converges uniformly to $g$, then $f$ is differentiable and $f' = g$ (i.e. $\lim f_n' = (\lim f_n)'$).

State the Weierstrass M-test for uniform convergence of $\sum f_n$.

If $|f_n(x)| \leq M_n$ for all $x \in E$ and $\sum M_n$ converges, then $\sum f_n$ converges uniformly (and absolutely) on $E$.

State the Cauchy criterion for uniform convergence of $(f_n)$.

$(f_n)$ converges uniformly on $E$ iff for every $\varepsilon > 0$ there is $N$ such that $\sup_{x\in E}|f_m(x) - f_n(x)| < \varepsilon$ for all $m,n \geq N$.

Give an example where pointwise convergence fails to preserve continuity.

$f_n(x) = x^{n}$ on $[0,1]$ converges pointwise to $f(x)=0$ for $x\in[0,1)$ and $f(1)=1$, a discontinuous limit; convergence is not uniform.

State the Comparison Test for series of nonnegative terms.

If $0 \leq a_n \leq b_n$: if $\sum b_n$ converges then $\sum a_n$ converges; if $\sum a_n$ diverges then $\sum b_n$ diverges.

State the Integral Test for convergence of a series.

If $f$ is positive, continuous, and decreasing on $[1,\infty)$ with $f(n)=a_n$, then $\sum_{n=1}^\infty a_n$ converges iff $\int_1^\infty f(x)\,dx$ converges.

Give the formula for the radius of convergence $R$ of a power series $\sum a_n x^n$.

$\frac{1}{R} = \limsup_{n\to\infty}\sqrt[n]{|a_n|}$ (Cauchy-Hadamard). The series converges absolutely for $|x| < R$ and diverges for $|x| > R$.

What is the Nested Interval Property and what completeness fact does it express?

If $I_n = [a_n,b_n]$ are closed bounded intervals with $I_1 \supseteq I_2 \supseteq \cdots$, then $\bigcap_n I_n \neq \emptyset$. If additionally $b_n - a_n \to 0$, the intersection is a single point. It is equivalent to completeness of $\mathbb{R}$.