RGU Question Paper Mathematics Semester IV 2021: Real Analysis-I (Old)

 June-2021

B.A/B. Sc. (IV Semester) Examination

MATHEMATICS

Paper: MATH-241 (OLD)

(Real Analysis-I)

Full Marks : 80

Pass Marks : 35%

Time : Three Hours

Notes: (i) Answer all questions.

(ii) The figures in the margin indicate full marks for the questions.

I. Answer any five parts: 𝟐 × 𝟓 = 𝟏𝟎

(a) If $a_n=\frac{(-1{)}^n}{n^2}, n\in \mathbb{N}$, find

(b) State Rolle’s theorem and give its geometrical interpretation.

(c) Prove that the set $S=\{1,\frac{1}{2},\frac{-1}{2},\frac{1}{3},\frac{-1}{3},...\}$ is neither open nor closed.

(d) Discuss the continuity of the function

(e) Give an example to show that intersection of arbitrary family of open sets is not open.

(f) Prove that a necessary condition for convergence of an infinite series $\sum u_n$ is that $\underset{n\to \infty }{\lim }{u}_n=0$.

(g) Prove that $\underset{x\to 0}{\lim }x\sin (1/x)=0$.

II. Answer any four parts: 𝟓 × 𝟒 = 𝟐𝟎

(a) Prove that a necessary and sufficient condition for the convergence of a monotonic function is that it is bounded.

(b) Show that

converges to zero.

(𝑖𝑖) 

(c) Evaluate

(d) Define uniform continuity. Show that the function 𝑓(𝑥) = 1/𝑥 is not uniformly continuous on (0,1].

(e) Show that

where |𝑥| < 1 and 𝑚 is any real number.

(f) Prove that the derived set of a bounded infinite set in ℝ has the smallest and greatest member.

(g) Prove that every absolutely convergent series is convergent.

III. Answer any five parts: 𝟏𝟎 × 𝟓 = 𝟓𝟎

(a) (i) Prove that every bounded sequence with a unique limit point is convergent. (5)

(ii) Prove that the sequence $\left\{r_n\right\}$ converges to zero if and only if |𝑟| < 1. (5)

(b)(i) If a function 𝑓 is continuous on [𝑎, 𝑏], then prove that it attains its bounds at least once in [𝑎, 𝑏]. (5)

(ii) Prove that the Dirichlet’s function 𝑓 defined on ℝ by

is discontinuous at every point. (5)

(c)(i)Prove that the intersection of an arbitrary family of closed set is closed. (5)

(ii) Prove that a countable union of countable sets is countable. (5)

(d) What is an alternating series? State and prove Leibnitz test for alternating series.

(e) (i) If a function 𝑓 is such that its derivative 𝑓′ is continuous on [𝑎, 𝑏] and derivable on (𝑎, 𝑏), then show that there exists a number 𝑐 between 𝑎 and 𝑏 such that $f\left(b\right)=f\left(a\right)+(b-a)f\text{'}\left(a\right)+\frac{1}{2}(b-a{)}^{2}f\text{'}\text{'}(c).$. (6)

[3]

(ii) Show that

(4)

(f) Test the convergence of the series

(5+5)

(g) Prove that a function which is derivable at a point is necessarily continuous. Show that continuity is not a sufficient condition for derivability using suitable example.