I Binomial, exponential theorems –their statements and proofs – their immediate
application to summation only. Logarithmic series theorem – statement and proof – immediate
application to summation only. Convergence and divergence of series – definitions, elementary
results- Comparison tests – De Alemberts and Cauchy’s tests.
II Absolute convergence- series of positive terms – Cauchy’s condensation test –Raabe’s
test. Theory of equations- roots of an equations – relations connecting the roots and coefficients .
Transformations of equations –character and position of roots – Descarte’s rule of signs –
symmetric functions of roots.
III Multiple roots – Rolle’s theorem – position of real roots of f(x) = 0. Newton’s method
of approximation to a root and Fourier’s rule – Horner’s method.
Curvature – radius of curvature in cartesian and polar forms – Evolutes and envelopes –
pedal equations – total differentiation – Euler’s theorem on homogeneous functions.
IV Integration of f’(x)/f(x) , ( px+q)/(ax2 + bx + c), √[(x-α) /(β-x)], √ [(x-α) (β-x)], 1/√
[(x-α) (β-x)], 1 /(acosx + bsinx +c), 1/(a cos2x + bsin2x +c) . Integration by parts. Reduction
formulae – problems – double and triple integrals –definitions – applications to calculations of
areas and volumes – area in polar coordinates.
Anx.18 D - B.Sc Maths (SDE) 2007-08 Page 3 of 26
V Approximate integration- Simpson’s rule and Trapezoidal rule, change of order of
integration in double integral- change of variables in double and triple integrals – Jacobians.
Notion of improper integrals- their convergence- simple tests for convergence- simple problems-
Beta and Gamma integrals – their properties-.relation between them.