Degree 4 Semester Maths questions Answers

REAL ANALYSIS

**UNIT-1: Sequences and Limits**

1. **State and prove the Sandwich Theorem (or Squeeze Theorem) for sequences.**

2. **State and prove Cauchy's First Theorem on Limits.**

3. **"Every convergent sequence is bounded." Is the converse true? Justify your answer.**

**UNIT-2: Series and Convergence Tests**

1. **State and prove the Leibniz Test for alternating series.**

2. **State and prove Cauchy's \(n\)-th Root Test for convergence of series.**

3. **State and prove D'Alembert's Test (or Ratio Test) for convergence of series.**

### **UNIT-3: Continuity and Functions**

1. **Discuss the continuity of the function \(f\) where \(f(x) = |x| + |x - 1|\) at \(x = 0\) and \(x = 1\).**

UNIT-4: Mean Value Theorems**

1. **State and prove Rolle's Theorem.**

2. **State and prove Lagrange's Mean Value Theorem (or First Mean Value Theorem).**

3. **State and prove Cauchy's Mean Value Theorem (or Second Mean Value Theorem).**

---

### **UNIT-5: Integral Calculus**

1. **State and prove the Fundamental Theorem of Integral Calculus.**

సమాధానాల కోసం " ఇక్కడ క్లిక్ "చెయ్యండి