Sequences and Series/Series and integration
Sequences and Series/Series and integration < Sequences and Series Jump to navigation Jump to search Theorem (Abelian partial summation) : Let ( � � ) � ∈ � be a sequence of complex numbers, and let � : [ 1 , ∞ ) → � be differentiable on ( 1 , ∞ ) . Finally define � ( � ) := ∑ 1 ≤ � ≤ � � � . Then for � ≥ 1 we have ∑ 1 ≤ � ≤ � � � � ( � ) = � ( � ) � ( � ) − ∫ 1 � � ( � ) � ′ ( � ) � � . Proof: If � = ⌊ � ⌋ , we have ∫ 1 � � ( � ) � ′ ( � ) � � = ∑ � = 2 � ∫ � − 1 � � ( � ) � ′ ( � ) � � + ∫ � � � ( � ) � ′ ( � ) � � = ∑ � = 2 � ∑ 1 ≤ � ≤ � − 1 � � ∫ � − 1 � � ′ ( � ) � � + ∑ 1 ≤ � ≤ � � � ∫ � � � ′ ( � ) � � = ∑ � = 2 � � ( � − 1 ) ( � ( � ) − � ( � − 1 ) ) + � ( � ) � ( � ) − � ( � ) � ( � ) . But ∑ � = 2 � � ( � − 1 ) ( � ( � ) − � ( � − 1 ) ) = ∑ � = 2 � � ( � − 1 ) � ( � ) − ∑ � = 2 � � ( � − 1 ) � ( � − 1 ) = ∑ � = 2 � � ( � − 1 ) � ( � ) − ∑ � = 1 � − 1 � ( � ) � ( � ) = ∑ � = 2 � − 1 � ( � ) ( � ( � − 1 ) − � ...