A test of convergence.

Dec 2009
1,506
434
Russia
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.
 
Dec 2009
411
131
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.

I'll think about this problem some more. Hopefully others have some more input for this problem.
 
Dec 2009
1,506
434
Russia
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.

I'll think about this problem some more. Hopefully others have some more input for this problem.


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?
 
Dec 2009
411
131
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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