# A test of convergence.

#### Also sprach Zarathustra

Let $$\displaystyle \sum_{k=1}^{\infty} b_k$$, $$\displaystyle \sum_{k=1}^{\infty} a_k$$ two series that $$\displaystyle a_k,b_k>0 \forall k \in \mathbb{N}$$.

Suppose that, $$\displaystyle \forall k \in \mathbb{N}$$ : $$\displaystyle a_k\leq|p(b_k)|$$, when $$\displaystyle p$$ is polynomial so that $$\displaystyle p(0)=0$$.
Prove that if $$\displaystyle \sum_{k=1}^{\infty} b_k$$ converges so then $$\displaystyle \sum_{k=1}^{\infty} a_k$$ converges also.

#### Dinkydoe

I'm not sure if this is of any help. But $$\displaystyle p(0) =0$$ we have $$\displaystyle p(x)=x\cdot q(x)$$ for some polynomial $$\displaystyle q$$

With the given condition we have: $$\displaystyle \sum_{k=1}^{\infty}a_k\leq \sum_{k=1}^{\infty}|p(b_k)|=\sum_{k=1}^{\infty}b_k\cdot|g(b_k)|$$

I myself, I'm not yet sure how to proceed from here. But I assume we like to conclude somehow that from $$\displaystyle \sum_{k=1}^{\infty}b_k$$ converges follows that
$$\displaystyle \sum_{k=1}^{\infty}b_k\cdot|g(b_k)|$$ converges.

#### Also sprach Zarathustra

I'm not sure if this is of any help. But $$\displaystyle p(0) =0$$ we have $$\displaystyle p(x)=x\cdot q(x)$$ for some polynomial $$\displaystyle q$$

With the given condition we have: $$\displaystyle \sum_{k=1}^{\infty}a_k\leq \sum_{k=1}^{\infty}|p(b_k)|=\sum_{k=1}^{\infty}b_k\cdot|g(b_k)|$$

I myself, I'm not yet sure how to proceed from here. But I assume we like to conclude somehow that from $$\displaystyle \sum_{k=1}^{\infty}b_k$$ converges follows that
$$\displaystyle \sum_{k=1}^{\infty}b_k\cdot|g(b_k)|$$ converges.

Maybe, using the fact that $$\displaystyle \lim_{k\to\infty}b_k=0$$, and from limit test:
$$\displaystyle \lim_{k\to\infty}\frac{b_kq(b_k)}{b_k}=0$$
We get the conclusion?

#### Dinkydoe

Ah, yes! That's it, the limit comparison test:

Since $$\displaystyle \lim_{k\to\infty}\frac{q(b_k)b_k}{b_k}= \lim_{k\to\infty}q(b_k)=(*) q(\lim_{k\to\infty}b_k)=q(0)$$

(* by continuity of q)

And $$\displaystyle q(0)$$ is some finite value since q is a polynomial. Therefore the series $$\displaystyle \sum_k b_k$$ and $$\displaystyle \sum_{k}q(b_k)b_k$$ must both converge or diverge. And there is your conclusion.

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