GATE Statistics Calculus Syllabus

Calculus syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Calculus in GATE Statistics, not a summary of it.

1. 1

   Finite, Countable and Uncountable Sets

   1 topic

   • Real Number System
2. 2

   Sequences of Real Numbers

   4 topics

   • Convergence of Sequences
   • Bounded Sequences
   • Monotonic Sequences
   • Cauchy Criterion for Convergence
3. 3

   Series of Real Numbers

   4 topics

   • Convergence
   • Tests of Convergence
   • Alternating Series
   • Absolute and Conditional Convergence
4. 4

   Power Series and Radius of Convergence

   overview

   Examined as a single unit within Calculus — no further topic split in the official outline.

5. 5

   Functions of a Real Variable

   12 topics

   • Limit
   • Continuity
   • Monotone Functions
   • Uniform Continuity
   • Differentiability
   • Rolle’s Theorem
   • Mean Value Theorems
   • Taylor’s Theorem
   • L’Hospital Rules
   • Maxima and Minima
   • Riemann Integration and Its Properties
   • Improper Integrals
6. 6

   Functions of Several Real Variables

   11 topics

   • Limit
   • Continuity
   • Partial Derivatives
   • Directional Derivatives
   • Gradient
   • Taylor’s Theorem
   • Total Derivative
   • Maxima and Minima
   • Saddle Point
   • Method of Lagrange Multipliers
   • Double and Triple Integrals and Their Applications

Calculus flashcards for GATE Statistics

25 of 50 cards from the Calculus deck — real questions with worked answers.

1. State the completeness (least upper bound) axiom of the real number system $\mathbb{R}$.

   Every non-empty subset of $\mathbb{R}$ that is bounded above has a least upper bound (supremum) in $\mathbb{R}$. Equivalently, every non-empty set bounded below has a greatest lower bound (infimum).

2. Define the supremum $\sup S$ of a set $S \subseteq \mathbb{R}$.

   $\alpha = \sup S$ is the least upper bound: (i) $x \leq \alpha$ for all $x \in S$, and (ii) for every $\varepsilon > 0$ there exists $x \in S$ with $x > \alpha - \varepsilon$.

3. State the Archimedean property of $\mathbb{R}$.

   For any real numbers $a > 0$ and $b$, there exists a natural number $n$ such that $na > b$. Equivalently, $\mathbb{N}$ is unbounded above in $\mathbb{R}$.

4. What does the density of rationals in $\mathbb{R}$ state?

   Between any two distinct real numbers $a < b$ there exists a rational number $q$ with $a < q < b$ (and also an irrational number).

5. Give the $\varepsilon$-$N$ definition of convergence of a sequence $(a_n)$ to limit $L$.

   $a_n \to L$ means: for every $\varepsilon > 0$ there exists $N \in \mathbb{N}$ such that $|a_n - L| < \varepsilon$ for all $n \geq N$.

6. Is a convergent sequence necessarily bounded? Is the converse true?

   Every convergent sequence is bounded. The converse is false: a bounded sequence need not converge, e.g. $a_n = (-1)^n$.

7. Define a bounded sequence $(a_n)$.

   $(a_n)$ is bounded if there exists $M > 0$ such that $|a_n| \leq M$ for all $n$; equivalently it is bounded above and below.

8. State the Bolzano–Weierstrass theorem for sequences.

   Every bounded sequence of real numbers has a convergent subsequence.

9. Define a monotonic (monotone) sequence.

   $(a_n)$ is monotonically increasing if $a_{n+1} \geq a_n$ for all $n$, and monotonically decreasing if $a_{n+1} \leq a_n$ for all $n$. It is monotonic if it is one of these.

10. State the Monotone Convergence Theorem for sequences.

    A monotonic sequence converges if and only if it is bounded. An increasing sequence bounded above converges to its supremum; a decreasing sequence bounded below converges to its infimum.

11. Define a Cauchy sequence $(a_n)$.

    $(a_n)$ is Cauchy if for every $\varepsilon > 0$ there exists $N$ such that $|a_m - a_n| < \varepsilon$ for all $m, n \geq N$.

12. State the Cauchy criterion for convergence in $\mathbb{R}$.

    A sequence of real numbers converges if and only if it is a Cauchy sequence. (This expresses the completeness of $\mathbb{R}$.)

13. What are $\limsup a_n$ and $\liminf a_n$, and when does $\lim a_n$ exist in terms of them?

    $\limsup a_n = \lim_{n\to\infty} \sup_{k \geq n} a_k$ and $\liminf a_n = \lim_{n\to\infty} \inf_{k \geq n} a_k$. The limit exists iff $\liminf a_n = \limsup a_n$.

14. State the Squeeze (Sandwich) theorem for sequences.

    If $a_n \leq b_n \leq c_n$ for all large $n$ and $a_n \to L$, $c_n \to L$, then $b_n \to L$.

15. For a series $\sum a_n$, what is the necessary condition (n-th term test) for convergence?

    If $\sum a_n$ converges then $a_n \to 0$. Conversely, if $a_n \not\to 0$ the series diverges. ($a_n \to 0$ alone is not sufficient.)

16. Define convergence of an infinite series $\sum_{n=1}^{\infty} a_n$.

    The series converges to sum $S$ if the sequence of partial sums $S_N = \sum_{n=1}^{N} a_n$ converges to $S$ as $N \to \infty$.

17. For the geometric series $\sum_{n=0}^{\infty} r^{n}$, when does it converge and to what?

    It converges iff $|r| < 1$, with sum $\dfrac{1}{1-r}$. It diverges for $|r| \geq 1$.

18. For the $p$-series $\sum_{n=1}^{\infty} \dfrac{1}{n^{p}}$, state the convergence condition.

    Converges if $p > 1$ and diverges if $p \leq 1$. In particular the harmonic series ($p=1$) diverges.

19. State the Comparison Test for positive-term series.

    If $0 \leq a_n \leq b_n$: convergence of $\sum b_n$ implies convergence of $\sum a_n$; divergence of $\sum a_n$ implies divergence of $\sum b_n$.

20. State the Limit Comparison Test.

    For $a_n, b_n > 0$, if $\lim_{n\to\infty} \dfrac{a_n}{b_n} = c$ with $0 < c < \infty$, then $\sum a_n$ and $\sum b_n$ both converge or both diverge.

21. 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.

22. 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.

23. 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.

24. 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.

25. 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 $1$.

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