Limit Comparison Test

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Theorem

Let $\left \langle {a_n} \right \rangle$ and $\left \langle {b_n} \right \rangle$ be sequences in $\R$.

Let $\displaystyle \frac {a_n}{b_n} \to l$ as $n \to \infty$ where $l \in \R_{>0}$.


Then the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ and $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ are either both convergent or both divergent.


Proof

Let $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ be convergent.

Then by Terms in Convergent Series Converge to Zero, $\left \langle {b_n} \right \rangle$ converges to zero.

A Convergent Sequence is Bounded.

So it follows that:

$\exists H: \forall n \in \N_{>0}: a_n \le H b_n$

Thus, by the corollary to the Comparison Test, $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is convergent.


Since $l > 0$, from Sequence Converges to Within Half Limit:

$\exists N: \forall n > N: a_n > \dfrac 1 2 l b_n$

Hence the convergence of $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ implies the convergence of $\displaystyle \sum_{n \mathop = 1}^\infty b_n$.

$\blacksquare$


Sources