3
$\begingroup$

Let $(a_n), (b_n), (c_n)$ be sequences of real numbers such that $\forall n \in \mathbb{N} \ \lvert a_n-b_n \rvert \leq c_n$. I want to show that if the series $\sum_{n=0}^{\infty} a_n$ and $\sum_{n=0}^{\infty} c_n$ converge, the series $\sum_{n=0}^{\infty} b_n$ also converges. Do you have any hints?

$\endgroup$
  • 5
    $\begingroup$ Hint: What do you know about each series in the RHS of the identity $$\sum b_n=\sum a_n+\sum (b_n-a_n)\ ?$$ $\endgroup$ – Did Dec 12 '16 at 20:43
2
$\begingroup$

Hint: you can use the completeness of real numbers. For $p\geq q$: \begin{aligned} \left|\sum_{n=0}^pb_n-\sum_{n=0}^qb_n\right|&=\left|\sum_{n=q+1}^pb_n\right|\\ &=\left|\sum_{n=q+1}^pa_n+\sum_{n=q+1}^p(a_n-b_n)\right|\\ &\leq \left|\sum_{n=q+1}^pa_n\right|+\sum_{n=q+1}^pc_n\\ &\leq \left|\sum_{n=q+1}^pa_n\right|+\left|\sum_{n=q+1}^pc_n\right|. \end{aligned} You can make the two expressions on the last line arbitrarily small by the Cauchy properties of the sequence of partial sums of $a_n$ and $c_n$, respectively.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.