Home › Previous year questions › GATE 2022 ST › Question 37GATE 2022 ST – Question 37 Calculus · Sequences and Series · 2 marks · Multiple choice
Which series converge conditionally? S₁=Σ_(n=0)^∞(−1)^n n/(n²+4); S₂=Σ_(n=0)^∞(−1)^n(√(n²+1)−n).
S₁ only S₂ only Both S₁ and S₂ Neither Practise this question in The GATE Grind →
Show answer and explanation Correct answer: (C) Both S₁ and S₂
Explanation Both positive term magnitudes eventually decrease to zero,so the alternating series converge. For S₁,n/(n²+4)∼1/n. Rationalizing S₂ gives 1/(√(n²+1)+n)∼1/(2n). Both absolute series diverge by limit comparison with the harmonic series. **C**.
More Sequences and Series questions GATE 2021 ST Q20 – Find n(2n+n2n2(n/2))1/(2n-n(1/n)). GATE 2026 ST Q37 – For x₁∈(0,4), define x (n+1)=2−√(4−x n). I:x n→0. II:x (n+1)/x n→1/4. Which hold? GATE 2025 ST Q12 – Which statement about countability is correct? GATE 2025 ST Q47 – For n≥1, x n=1+1/√2+1/√3+⋯+1/√n−2(√n−1). Which hold? GATE 2024 ST Q12 – Let a₁=√6 and a (n+1)=√(6+a n). I:the sequence increases. II:its limit is 2. Which hold? GATE 2024 ST Q30 – For Σₙ₌₀∞a nxⁿ, coefficients satisfy a (2n+1)=1/2(2n+1) and a (2n)=1/3(2n),n≥0. Find its… All Sequences and Series questions →