Mathematics Paper-I REAL ANALYSIS (वास्तविक विश्लेषण ) Book B.Sc 3rd Sem Bihar

Unit

Subject

No. of Lectures

1

Dedekind theory of real numbers, Algebraic and order properties of R, Archimedean Property, Density Theorem, Completeness property of R, Bounded sets, Theorems on Suprema and lnfima.

10

2

Neighbourhood of a point in R, Open and closed sets, Limit points and isolated points of a set, Bolzano-Weierstrass theorem for a set, Derived set, Clouser and Interior of a set.

12

3

Sequence and its convergence, Bounded sequence, Monotone sequences, Subsequences, Limit of a sequence, Limit Theorem, Bolzano-Weierstrass theorem for sequences, Limit superior and limit inferior for bounded sequence, Cauchy sequence, Cauchy’s general principle of convergence.

14

4

Infinite series and their convergence, Cauchy Criterion, Tests for convergence: Comparison test, D’Alembert Ratio Test, Cauchy's root test, Rabbe’s test, Logarithmic test, D’Morgan and Bertrand’s test, Cauchy integral test, Cauchy condensation test, Gauss’s test, Alternating series, Leibnitz test, Absolute and Conditional convergence.

14

bdkbZ

fo"k;

No. of Lectures

1

okLrfod la[;kvksa dk MsMsdkbaM fl)kar] chtxf.krh; rFkk R ds Øe xq.k] vkfdZfeMh; xq.k] ?kuRo çes;] R dk iw.kZrk Áxq.k] ifjc) leqPp;] lqçsek vkSj bfUQek ij çes;

10

2

R esa ,d fcanq dk lkehI;] foo`r rFkk lao`r leqPp;] lhek fcanq rFkk i`Fkd~ fcUnq dk ,d leqPp;]

,d leqPp; ds fy, cksYt+kuks&oh;jLVªSl çes;, O;qRiUu leqPp;] ,d leqPp; dk laojd rFkk vkarfjd

12

3

vuqØe rFkk vfHklkfjrk] ifjc) vuqØe] ,dfn"Vrk vuqØe] mivuqØe] vuqØe dh lhek] lhek izes;] vuqØeksa ds fy, cksYt+kuks& oh;jLVªSl çes;] ifjc) vuqØe dh mPp lhek ,oa fuEu lhek] dkS’kh vuqØe] dkS’kh ds vfHklkfjrk dk lkekU; fl)kar

14

4

vifjfer Js.kh rFkk mudh vfHklfjrk] dkS’kh ekunaM] vfHklfjrk ds fy, ijh{k.k& rqyuk ijh{k.k] 

D’ ,ysEcVZ vuqikr ijh{k.k] dkS’kh dk ewy ijh{k.k] jkcs dk ijh{k.k] y?kqx.kdh; ijh{k.k] D’ e‚xZu vkSj cVªsZaM ijh{k.k] dkS’kh lekdy ijh{k.k] dkS’kh la?kuu ijh{k.k] x‚l dk ijh{k.k] ,dkUrj Js.kh]

fyCuht ijh{k.k] iw.kZ ,oa l'krZ vfHklkfjrk

14

Unit

Subject

No. of Lectures

1

Dedekind theory of real numbers, Algebraic and order properties of R, Archimedean Property, Density Theorem, Completeness property of R, Bounded sets, Theorems on Suprema and Infima.

10

2

Sequence and its convergence, Bounded sequence, Monotone sequences, Subsequences, Limit of a sequence, Limit Theorem, Bolzano-Weierstrass theorem for sequences, Cauchy sequence, Cauchy’s general principle of convergence.

10

3

Infinite series and their convergence, Cauchy Criterion, Tests for convergence: Comparison test, D’Alembert Ratio Test, Cauchy’s root test, Rabbe’s test, Logarithmic test, Cauchy integral test, Gauss’s test, Alternating series, Leibnitz test, Absolute and Conditional convergence.

10

bdkbZ

fo"k;

No. of Lectures

1

okLrfod la[;kvksa dk MsMsdkbaM fl)kar] chtxf.krh; rFkk R ds Øe xq.k] vkfdZfeMh; xq.k] ?kuRo çes;] R dk iw.kZrk Áxq.k] ifjc) leqPp;] lqçsek vkSj bfUQek ij çes;

10

2

vuqØe rFkk vfHklkfjrk] ifjc) vuqØe] ,dfn"Vrk vuqØe] mivuqØe] vuqØe dh lhek] lhek izes;] vuqØeksa ds fy, cksYt+kuks& oh;jLVªSl çes;] dkS’kh vuqØe] dkS’kh ds vfHklkfjrk dk lkekU; fl)kar

10

3

vifjfer Js.kh rFkk mudh vfHklfjrk] dkS’kh ekunaM] vfHklfjrk ds fy, ijh{k.k& rqyuk ijh{k.k]  D’ ,ysEcVZ vuqikr ijh{k.k] dkS’kh dk ewy ijh{k.k] jkcs dk ijh{k.k] y?kqx.kdh; ijh{k.k] dkS’kh lekdy ijh{k.k] x‚l dk ijh{k.k] ,dkUrj Js.kh] fyCuht ijh{k.k] iw.kZ ,oa l'krZ vfHklkfjrk

10