The GATE Grind

GATE 2023 ST – Question 50

Calculus · Sequences and Series · 2 marks · Multiple select

For real a define a⁺=max(a,0),a⁻=max(−a,0). Which statements about real sequences are true?

  1. Convergence of x_n implies convergence of x_n⁺ and x_n⁻.
  2. x_n→0 implies x_n⁺→0 and x_n⁻→0.
  3. Convergence of both x_n⁺ and x_n⁻ implies convergence of x_n.
  4. Convergence of x_n² implies convergence of both x_n⁺ and x_n⁻.

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) Convergence of x_n implies convergence of x_n⁺ and x_n⁻.; (B) x_n→0 implies x_n⁺→0 and x_n⁻→0.; (C) Convergence of both x_n⁺ and x_n⁻ implies convergence of x_n.

Explanation

The positive- and negative-part maps are continuous,proving A and B. Since x_n=x_n⁺−x_n⁻,C follows from limit algebra. D fails for x_n=(−1)^n:the squares equal 1 but both parts alternate 0,1. **A,B,C**.

More Sequences and Series questions

All Sequences and Series questions →