(a) Using Cauchy's general principle of convergence, examine the convergence of the sequence , where fₙ = 1 + 1/1! + 1/2! + ... + 1/n!. 15 marks (b) Show that every homomorphic image of an abelian group is abelian, but the converse is not nec

Question

(a) Using Cauchy's general principle of convergence, examine the convergence of the sequence < fₙ >, where fₙ = 1 + 1/1! + 1/2! + ... + 1/n!.

15 marks

(b) Show that every homomorphic image of an abelian group is abelian, but the converse is not necessarily true.

15 marks

(c) Find the function which is analytic inside and on the circle C : z = e^(iθ), 0 ≤ θ ≤ 2π and has the value

(a² - 1) cos θ + i(a² + 1) sin θ
──────────────────────────────
a⁴ - 2a² cos 2θ + 1

on the circumference of C, where a² > 1.

20 marks

UPSC Answer Check · Accepted Answer

Prove the three mathematical statements systematically: for (a) apply Cauchy's general principle by showing |f_{n+p} - f_n| < ε for n ≥ m; for (b) prove the homomorphism property for abelian groups and provide a concrete counterexample for the converse; for (c) use Poisson's integral formula or Schwarz's formula to construct the analytic function from boundary values. Allocate approximately 30% time to (a), 30% to (b), and 40% to (c) given their mark distribution and complexity.
- For (a): Correct application of Cauchy's general principle showing |f_{n+p} - f_n| = Σ_{k=n+1}^{n+p} 1/k! < 1/n!·(e-1) → 0 as n → ∞, establishing convergence to e
- For (b): Proof that φ(ab) = φ(a)φ(b) = φ(b)φ(a) = φ(ba) for homomorphism φ: G → H when G is abelian; counterexample using S₃ (non-abelian) with abelian quotient S₃/A₃ ≅ ℤ₂
- For (c): Recognition that boundary data represents real part of analytic function; application of Schwarz's formula or Poisson integral to find harmonic conjugate
- For (c): Simplification of denominator using a⁴ - 2a²cos2θ + 1 = |a² - e^{2iθ}|² and identification with Re[(a²+z²)/(a²-z²)] or similar standard form
- For (c): Final analytic function f(z) = (a²+z²)/(a²-z²) or equivalent verified on |z|=1
- Clear logical flow with proper mathematical notation and statement of theorems used

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	18%	9	For (a): correctly identifies sequence as partial sums of e and sets up Cauchy criterion with proper ε-N structure; for (b): clearly defines homomorphism and states what needs to be proved; for (c): correctly interprets boundary condition as real/imaginary parts and identifies C as unit circle	Basic setup present but missing some definitions or misidentifies the nature of one sub-part (e.g., confuses sequence with series in (a), or fails to specify domain of analyticity in (c))	Major setup errors: wrong theorem identified for (a), confuses kernel with image in (b), or fails to recognize the circle parameterization in (c)
Method choice	22%	11	For (a): uses comparison with geometric series or direct factorial bound; for (b): employs fundamental homomorphism theorem structure with clean counterexample selection; for (c): selects Schwarz's formula/Poisson kernel or contour integration method appropriate for Dirichlet problem	Correct general approach but suboptimal method (e.g., attempts ratio test for (a) instead of Cauchy principle, or uses brute force verification rather than structural proof for (b))	Wrong method entirely (e.g., uses root test for (a), attempts Lagrange's theorem for abelian proof in (b), or tries Taylor series expansion without boundary matching in (c))
Computation accuracy	22%	11	For (a): precise inequality chain showing remainder < 1/n·n! or similar sharp bound; for (b): explicit computation of φ(ab) and valid counterexample verification; for (c): correct algebraic manipulation of trigonometric identities and complex arithmetic leading to f(z)=(a²+z²)/(a²-z²)	Correct final answers but with minor computational slips (e.g., arithmetic errors in factorial bounds, sign errors in group elements, or coefficient mistakes in partial fractions)	Serious computational errors: incorrect remainder estimation in (a), invalid group operation in counterexample for (b), or failure to simplify (a²-1)cosθ + i(a²+1)sinθ correctly in (c)
Step justification	22%	11	Every logical step explicitly justified: ε-N definition properly invoked in (a), homomorphism properties clearly cited in (b), uniqueness of analytic continuation and maximum modulus principle referenced in (c); cites 'Cauchy's General Principle of Convergence' and 'First Isomorphism Theorem' by name	Correct reasoning present but gaps in explicit justification (e.g., assumes convergence of factorial series without remark, skips verification that kernel is normal in counterexample, asserts analyticity without checking)	Unjustified leaps: claims 'clearly converges' without ε argument in (a), asserts 'hence abelian' without showing commutativity in (b), or states answer without deriving from boundary data in (c)
Final answer & units	16%	8	For (a): concludes sequence converges to e (or 'converges' with limit identified); for (b): complete statement with both directions addressed; for (c): explicit formula f(z)=(a²+z²)/(a²-z²) or equivalent, verified on \|z\|=1; all parts clearly labeled and boxed	Correct conclusions but poorly presented (e.g., missing limit identification in (a), incomplete counterexample description in (b), or correct function without verification in (c))	Missing or wrong conclusions: no convergence statement for (a), missing converse direction in (b), or no function found for (c); or presents numerical approximation instead of exact form

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