Science:Infinite Series Module/Units/Unit 2/2.1 The Divergence Test/2.1.02 A Useful Theorem
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
Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align} s_n = \sum_{k=1}^{n} a_k = a_1 + a_2 + a_3 + \ldots +a_{n1} + a_n.\end{align} }
Then
and
By assumption, a_{n} is convergent, so the sequence {s_{n}} is convergent (using the definition of a convergent infinite series). Let the number S be given by
Since n1 also tends to infinity as n tends to infinity, we also have
Finally,
Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align} \lim_{n \rightarrow \infty} a_n &= \lim_{n \rightarrow \infty} (s_n  s_{n1}) \\ &= S  S \\ &=0. \end{align}}
Thus, if
is convergent, then
as required.