課程名稱︰分析導論優一
課程性質︰數學系大二必修
課程教師︰王振男
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2014/12/30
考試時限（分鐘）：40
試題 :
n
1. (10%) Let f (x) = cos x if 0 ≦ x ≦ π.
n
(a) Prove that {f } converges in the mean to 0 on [0, π] but that {f (π)}
n n
does not converge.
(b) Prove that {f } converges pointwise but not uniformly on [0, π/2].
n
∞ n
2. (10%) Given the power series Σ a z has radius of convergence 2. Find the
n=0 n
radius of convergence of each of the following series:
2
∞ k n ∞ kn ∞ n
Σ a z , Σ a z , Σ a z ,
n=0 n n=0 n n=0 n
where k is a fixed positive integer.