|
|||||
A sequence whose terms alternate in sign is called an alternating
sequence, and such a sequence converges if two simple conditions hold, Why are these conditions sufficient for convergence? (a1 + a2) + (a3 + a4) + (a5 + a6) + (a7 + a8) + ... and a1 + (a2 + a3) + (a4 + a5) + (a6 + a7) + (a8 + a9) + ... Notice that because the series is alternating in sign, the terms in parentheses
are differences of the absolute values of successive terms, and by our first condition
they all have the signs of their first terms. Suppose, for example, that a1 is
positive. Then all the terms in the first expression are positive, and its partial
sums will increase. On the other hand, all the terms in the second expression
after the first will be negative, and its partial sums will decrease. Thus, the
even partial sums of our series will increase and the odd partial sums will decrease.
The value of all later partial sums must therefore at each stage lie sandwiched
between them. By our second condition differences in successive partial sums (which
are the terms in the series themselves) approach 0, which means that the values
of the partial sums are constricted to intervals that approach 0 in size, which
implies that the partial sums converge in the sense of Cauchy (their differences
approach 0.) Exercise 30.3 Use a spreadsheet to find the first 40 partial sums of the
alternating harmonic sequence. Then take the sum of successive pairs and repeat
20 times. (this can be done with one instruction of the form ,
copied into columns d, e, f, g, ...) |