A Useful Theorem

Previous: Introduction to the Divergence Test

Next: The Divergence Test




The following theorem will yield the divergence test.

Theorem 1

If the infinite series

is convergent, then

Proof of Theorem 1

The proof of this theorem can be found in most introductory calculus textbooks that cover the divergence test and is supplied here for convenience. Let the partial sum be

Then

and

By assumption, an is convergent, so the sequence {sn} is convergent (using the definition of a convergent infinite series). Let the number S be given by

Since n-1 also tends to infinity as n tends to infinity, we also have

Finally,

Thus, if

is convergent, then

as required.


Previous: Introduction to the Divergence Test

Next: The Divergence Test




Leave a Reply

Your email address will not be published. Required fields are marked *