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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI 600 034
M.Sc.DEGREE EXAMINATION STATISTICS
FIRSTSEMESTER APRIL 2018
17/16PST1MC03- STATISTICAL MATHEMATICS
Date: 28-04-2018 Dept. No. Max. 100 Marks
Time: 09:00-12:00
SECTION A
Answer ALL questions. Each carries TWO marks. (10 x 2 20 marks)
1. Prove that a divergent sequence may have a convergent subsequence.
2. Prove that the series 1 2 3 ... n ... is divergent.
3. If L is the limit of the sequence then prove that every open interval containing L
contains all but a finite number of terms of the sequence.
4. Show that all subsequences of a convergent sequence of real numbers converge to the same
limit.
5. Write the different formulae for differentiating the sum, difference, product and quotient of
two functions.
6. State comparison test for the series of positive terms.
7. Show that the function x2 is unbounded on R but is bounded on each bounded interval
of R.
8. Let for x ∞). Prove that does not have a derivative at even though
is continuous at
9. Define upper and lower integral of a bounded function f over b].
10. Explain linear independence of k vectors using an example.
SECTION B
Answer any FIVE questions. Each carries EIGHT marks. x 8 40 marks)
11. Write the formula for sn for the sequence 13, ... . Verify if the following
sequence is a subsequence of
13, ...
...)
13, ...)
10, 16, ...).
12. State and establish Cauchy Criterion of Convergence of a series.
13. If the limit of a sequence of real numbers exists, then prove that it is unique.
14. Let Σ be a series of non-negative numbers and let sn a1 a2 ... an. Then show that
Σ if is bounded.
2
if is not bounded.
15. If converges absolutely, then prove that the series converges but not conversely.
16. Prove that each constant function c is Riemann integrable on any interval for
any partition P of b].
17. Let x x 1). Let σ be the partition 14, 24, 34, of 1]. Obtain U
and σ].
18. Mention any four properties of the Riemann integral.
SECTION C
Answer any TWO questions. Each carries TWENTY marks. x 20 40 marks)
19(a). Show that a monotonic series converges if and only if it is bounded.
19(b). Obtain → where c is a fixed positive number.
20. State and establish Leibnitz Rule on alternating series.
21(a). Explain improper integral of the first, second and third kind and give an example for each
kind.
21(b). Check the convergence of the integrals:

22(a). State Taylor's formula and Maclaurin's Theorem with Lagranges form of remainder.
Hence write Taylor's formula for log(1 about a 2 and n 4.
22(b). Explain the inductive procedure of Gram-Schmidt Orthogonalization.