Absolute Convergence

Kept Girl - Absolutely Convergence Dorothy Moorefield May 5, 2009

Recap The goal of these few write-ups is to understand the relation: Y p

∞ X 1 1 = . 1 ns 1 − ps n=1

Which lead to this theorem: P f (n) is absolutely Theorem 0.1 Let f be a multiplicative arithmetical function such that the series convergent. Then the sum of the series can be expressed as an absolutely convergent infinite product, ∞ X

f (n) =

Y [1 + f (p) + f (p2 ) + . . . ] p

n=1

extended over all primes. If f is completely multiplicative, the product simplifies and we have ∞ X n=1

f (n) =

Y p

1 . 1 − f (p)

Which lead to the definitions of multiplicative and completely multiplicative arithmetical functions as well as absolute convergence. Recall: Definition 0.2 A real or complex-values function defined on the positive integers is called an arithmetical function. Definition 0.3 An arithmetical function f is called multiplicative if f is not identically zero and if f (mn) = f (m)f (n) whenever (m, n) = 1. A multiplicative function f is called completely multiplicative if we also have f (mn) = f (m)f (n) for all m, n. For more details on these definitions please see previous posts. absolute convergence Now we go back to Lang’s book ”Complex Analysis”. Let {zn }∞ n=1 be a sequence of complex numbers. 1

2

Kept Girl

Definition 0.4 Consider the partial sum: sn =

n X

zk .

k=1

The we say the series,

∞ X

zk

k=1

converges if lim sn = w

n→∞

for some w ∈ C. We should discuss lim sn = w.

n→∞

This is read as ”the limit and n goes to infinity of sn is w. The notion of limits is very important when dealing with any type of analysis. It is possible to get through calculus without really understanding limits, however, if you did so, then I’d say you did not understand calculus at all. I am not going to go into a full-blown explanation of limits. If you are uncomfortable with limits, I highly encourage you to pick up a calc book and get friendly with them. Limits are your friends!

Definition 0.5 Let

∞ X

an

n=1

be a series of complex numbers. Then the series converges absolutely if the real positive series ∞ X

|an |

n=1

converges. There are a few ways to check if a series converges without actually computing what it converges too. When we wrap all these definitions up to use our main theorem to show our desired equality we will use one of these tests. I have yet to work it out but I suspect we will use: P Theorem 0.6 Let P xn be a series of nonnegative real numbers, which converges. If |an | ≤ xn for all n, then the series an converges absolutely.