MTH603 Midterm MCQs 2024 by Miss Mehwish

Past Papers for Mids./Finals are also Available · MTH603 (Mega File) · Mid Term (Live Quiz)

1. For the power series, define R in the extended real numbers by 1/R = lim sup n→∞ |aₙ |¹/n. In particular, R = ∞ if... lim sup n→∞ |aₙ |¹/n = 0.

lim sup n→∞ |aₙ |¹/n = ∞. ✅

lim sup n→∞ |aₙ |¹/n = 1. lim sup n→∞ |aₙ |¹/n = -∞.

2. For the power series; f(z) = ∑ (−1)ⁿ z^(2n+1) / (2n+1)!, then f'(z) =...
3. tan x is an odd function, its derivatives of even order are equal to... 1 at x₀ = 0. π at x₀ = 0.

0 at x₀ = 0. ✅

tan x sec x at x₀ = 0.

4. The power series expansion of f(x) = sin x about x₀ = 0 is... ∑∞ n=0 x²ⁿ / (2n)!

∑∞ n=0 x²ⁿ⁺¹ / (2n+1)!

5. If f(x) = sin x, then...

f''(x) = sin x ✅

f²ᵐ⁺¹(0) = 1, m ≥ 0. f²ᵐ(0) = 1. ||f^(k)||(-∞,∞) = 1, k ≥ 0.

6. If a function f(x), having the derivatives of all orders on some neighborhood N(0) of 0, be chosen first and the Taylor series ∑∞ n=0 f^(n)(0) / n! xⁿ be constructed, then this power series will have f(x) as its sum on N(0). True ✅ False
7. Let f be defined by a power series with finite radius of convergence R. Then lim x→(x₀+R)− f(x) = ∑∞ n=0 aₙ Rⁿ only if...

∑∞ n=0 aₙ Rⁿ converges. ✅

∑∞ n=0 bₙ Rⁿ is uniformly convergent.