Science:Infinite Series Module/Units/Unit 2/2.1 The Divergence Test/2.1.02 A Useful Theorem

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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_{n-1} +  a_n.\end{align} }

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,

Failed to parse (unknown function "\begin{align}"): {\displaystyle \begin{align} \lim_{n \rightarrow \infty} a_n &= \lim_{n \rightarrow \infty} (s_n - s_{n-1}) \\  &= S - S \\ &=0. \end{align}}

Thus, if

is convergent, then

as required.