Sequences and Series

In Chapter 17 a function / of variable x, denoted by /(x), was considered. If x can only take integral values, generally denoted as n, then f n) defines a sequence. For example, if 

The sum of the terms of a sequence, is said to define a series. To determine the sum of the terms above we could take 

This is cumbersome to write, so we use mathematical shorthand and express the sum as 

Here S is the symbol for the Greek letter sigma, and simply means take the sum of whatever follows, in this case the general term /(n). The n = 1 below E means start at this value while the N above denotes the final value of n. 

There are certain series for which it is possible to determine precise formulae for their overall sums. One such example that is of interest to chemists is 

A sequence is a function that maps integers to a subset of real numbers. A sequence can be viewed as a set of numbers indexed by integers. A finite sequence terminates after a finite number of terms. The following is an example of a finite sequence  

A sequence that extends indefinitely is an infinite sequence. For example, a, = 2i+l Jo is 

If an infinite sequence approaches a definite number, then the sequence is convergent. Thus the real sequence represented above is a convergent sequence that converges to zero. 

A sequence is called increasing (decreasing) if a +i a a +i a ) and is called nondecreasing (nonincreasing) if a +i aSa + n)- A seqnence is called strictly monotone if it is either increasing or decreasing. A seqnence is bonnded if there is a nnmber M such that a M. A bounded strictly monotone sequence converges. 

A series is the snm of a seqnence. Thus, given a sequence, one can form a series by snmming the sequence. A finite seqnence generates a finite series, and an infinite sequence generates an infinite series. 

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Much of this chapter is concerned with a discussion of power series, but before we go into detail we consider the general concepts of sequences and series, both finite and infinite. 

Jeener s idea was to introduce an incremented time ti into the basic ID NMR pulse sequence and to record a series of experiments at different values of second dimension to NMR spectroscopy. Jeener described a novel experiment in which a coupled spin system is excited by a sequence of two pulses separated by a variable time interval <]. During these variable intervals, the spin system is allowed to evolve to different extents. This variable time is therefore termed the evolution time. The insertion of a variable time period between two pulses represents the prime feature distinguishing 2D NMR experiments from ID NMR experiments. 

Phenylthiocarbamoyl derivatives of 18 chiral amino acids were separated on a C8 column connected in series to a phenylcarbamoylated (3-cyclodextrin column (Iida et al., 1997). The Cg column separated the derivatized amino acids from one another entering the chiral column. Under this configuration several enantiomers of adjacent amino acids coeluted resulting in poor resolution. However, this configuration was successful in determining the amino acid sequence and chirality of the amino acids in a D-amino acid containing peptide. 

In this study, the effects of the variations in block sequence and composition (and thus relative block length) on the material properties of two series of triblock copolymers has been investigated. One of the blocks, the hydrogenated polybutadiene (HB), is semicrystalline, and the other block, the hydrogenated polyisoprene (HI) is rubbery at room temperature. Thus in one series, the HBIB block copolymers, the end blocks are semi-... 

Taylor, W. R. (1992) Patterns in Protein Sequence and Structure, Springer Series in Biophysics (Springer, Berlin), Vol. 7. 

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