Real Sequences : Can some terms be undefined?

[Legendre]

The definition of a sequence of real numbers is : a function from N to R.

What is the standard way to handle a sequence like (1/log n) where the first term is undefined? Do we instead write (1/log (n+1)) so that the first term is defined? Or leave the first term undefined?

The definition says that the function must map from N to R. So every element in N needs to have a valid image. Would undefined terms violate this definition? E.g. f(1) = (1/log 1) do not exists.


What happens when we have the sequence (An/Bn) where (An) and (Bn) are two convergent sequences with non-zero limits? Some of the terms of (An/Bn) might be undefined due to Bn being 0. Do we drop just those terms or drop the initial terms until all remaining terms are defined?

Thanks!

 

[HallsofIvy]

Yes, undefined terms are not members of R. But, yes, we can just drop any "undefined" terms from a sequence or drop initial terms until there are no more "undefined" terms. It will not change the limit. You can drop any finite number of terms from a sequence without changing the limit.

 

[Legendre]

Thank you! But...

1) Can you post the source of your information? (textbook etc) I looked through a few intro to analysis text but didn't come across the treatment of undefined terms.

2) Suppose we define a series using the sequence of the form (An/Bn) where (An) and (Bn) are sequences and the limit of (Bn) is not zero.

If some An/Bn terms are undefined then the sum of the series would depend on whether we drop initial terms or just drop the undefined terms.

But which is the "standard way" of doing it?

 

[g_edgar]

Whether you can "drop" terms depends on what you want to do. For example, to do a limit, the first few terms do not matter.

 

[Tac-Tics]

1) Can you post the source of your information? (textbook etc) I looked through a few intro to analysis text but didn't come across the treatment of undefined terms.

"Undefined" is an informal term. Strictly speaking, a limit cannot be defined by (1/log n) for n = 0, 1, 2, ... because log isn't defined at 0.

However, as Halls said, you can drop any finite number of terms from a series (your "undefined" terms) and the series maintains its limit.

The reason for this comes from the definition of a limit.

A number L is the limit of a sequence (s_i) iff for all real e > 0, you can find a natural number n such that when i > n, |s_i - L| < e.

If you drop anyone term from the sequence, you simply have to choose a value one greater for n. And because you can drop one term, you can repeat the process a finite number of times to get the same results.