Math253 absolute convergence, the ratio test, and power series PDF

Preview

Summary

Download Math253 absolute convergence, the ratio test, and power series PDF

Description

Math 253 – absolute convergence, the ratio test, and power series Absolute convergence of an infinite series simply means that the associated series of the absolute values of each term converges: 

a

 n

is absolutely convergent when

n= 1

a

n

is convergent.

n =1

For a series of all positive terms, both definitions are equivalent. It is not hard to show that any series which is absolutely convergent is also convergent, but the converse is not true. If a series is convergent but not absolutely convergent, we call it conditionally convergent. A good example of a conditionally convergent series is the alternating series derived from the harmonic series: 

 (−1) n= 1

n+ 1

1 1 1 1 1 = − + − + n 1 2 3 4

is easily proven to be convergent by the alternating series test, but taking the absolute value of each term gives us the harmonic series which we have proven to be divergent by a couple of different ways. One interesting aspect of absolutely convergent series is that the terms can be reordered without changing the limit value which the same thing is not true of conditionally convergent series. The Ratio Test is a very powerful convergence test which will be useful when we study 

power series. Given a series

a

n

n =1

, the Ratio Test says that if the lim a n+ 1 exists and is equal to n→

an

some (non-negative) real number L, then the series converges if L  1 and diverges if L  1. It also diverges if the limit does not exist, but if L = 1 , the Ratio Test is inconclusive and does not tell whether the series converges or diverges. In some cases, we can still determine whether the series converges or diverges by some other test such as the Integral Test or one of the comparison test....