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Algebraic and order properties of ℝ, Absolute value of a real number, Bounded above and
bounded below sets, Supremum and infimum of a non-empty subset of ℝ, The completeness
property of ℝ, Archimedean property, Density of rational numbers in ℝ.
UNIT - 2 Sequences (15 Hours)

Sequences and their limits, Convergent sequence, Limit theorems, Monotone sequences,
Monotone convergence theorem, Subsequence‘s, Bolzano-Weierstrass theorem for sequences,
Limit superior and limit inferior for bounded sequence, Cauchy sequence, Cauchy‘s convergence
criterion.
UNIT -3 Infinite Series (18 Hours)

Convergence and divergence of infinite series of real numbers, Necessary condition for convergence,
Cauchy criterion for convergence, Tests for convergence of positive term series, Integral test,
comparison test, D‘Alembert‘s ratio test, Cauchy‘s nth root test, Raabe‘s test, Alternating series,
Leibniz test, Absolute and conditional convergence.
Sequencesandseriesoffunctions,Pointwiseanduniformconvergence,Mn-test,M- test,Statementsof
theresultsaboutuniformconvergenceandintegrabilityand
differentiabilityoffunctions,Powerseriesandradiusofconvergence.

*TUTORIAL (15 Hours (1 Hour per week))

SUGGUESTED READINGS:

1. Bartle, Robert G., & Sherbert, Donald R. (2011). Introduction to Real Analysis (4th ed.).
John Wiley & Sons. Wiley India Edition 2015.
2. Bilodeau, Gerald G., Thie, Paul R., & Keough, G. E. (2010). An Introduction to Analysis
(2nd ed.). Jones and Bartlett India Pvt. Ltd. Student Edition. Reprinted 2015.
3. Denlinger, Charles G. (2011). Elements of Real Analysis. Jones and Bartlett India Pvt.
Ltd. Student Edition. Reprinted 2015.
4. Aliprantis C. D., & Burkinshaw, O. (1998). Principles of Real Analysis (3rd ed.).
Academic Press.
5. Ross, Kenneth A. (2013). Elementary Analysis: The Theory of Calculus (2nd ed.).
Undergraduate Texts in Mathematics, Springer. Indian reprint.
6. Thomson, B. S., Bruckner, A. M., & Bruckner, J. B. (2001). Elementary Real Analysis.
Prentice Hall.
7. Spectrum-Real Analysis, Sharma Publications, Jalandhar
8. A Text Book of Real Analysis, S. Dinesh & Co, Jalandhar
9. Spectrum-Sequences and Series, Sharma Publications, Jalandhar
10. A Text Book of Sequences and Series, S. Dinesh & Co, Jalandhar

Maths.221 Algebra 3+1*

LEARNING OBJECTIVES:

The primary objective of the course is to introduce:
 Modular arithmetic, fundamental theory of groups, rings, integral domains, and fields.
 Symmetry group of a plane figure, and basic concepts of cyclic groups.
 Cosets of a group and its properties, Lagrange‘s theorem, and quotient groups.

LEARNING OUTCOMES:

This course will enable the students to:
 Appreciate ample types of groups present around us which explains our surrounding

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