Posts

Sequences and Series/Series and integration

Image
  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 ) − � ...
Post a Comment
Read more

Sequence

Image
  Sequence 73 languages Article Talk Read Edit View history Tools From Wikipedia, the free encyclopedia "Sequential" redirects here. For the manual transmission, see  Sequential manual transmission . For the sequentional logic function, see  Sequention . For other uses, see  Sequence (disambiguation) . In  mathematics , a  sequence  is an enumerated collection of  objects  in which repetitions are allowed and  order  matters. Like a  set , it contains  members  (also called  elements , or  terms ). The number of elements (possibly  infinite ) is called the  length  of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a  function  from  natural numbers  (the positions of elements in the sequence) to the elements at each position. The not...
Post a Comment
Read more