# Thread: sequences

1. ## sequences

Hi all,
I'm trying to solve the following 2 questions , can anyone help?
1. Let for k . Show that there exists a sequence 0 such that

2.Let .Show that there exists a sequence such that .

For the first part , I think for is suitable. Can anyone give a hint for the second part?

2. If $\displaystyle \sum_{k=1}^{\infty} |a_{k}|< \infty$, then the series converges absolutely and that means that $a_{k} \rightarrow 0$ and for k 'large enough' is $\displaystyle |a_{k}|< \frac{1}{k^{\alpha}}$ where $\alpha>1$. Now if we suppose that $\forall k, a_{k}\ne 0$ and set $b_{k}= \ln |a_{k}|$ it will be $|b_{k}| \rightarrow \infty$ and for k 'large enough' $\displaystyle |a_{k}\ b_{k}|< \frac{1}{k^{\beta}}$ where $\beta>1$ so that $\displaystyle \sum_{k=1}^{\infty} |a_{k}\ b_{k}|< \infty$ ...

Kind regards

$\chi$ $\sigma$

3. Originally Posted by chisigma
If $\displaystyle \sum_{k=1}^{\infty} |a_{k}|< \infty$, then the series converges absolutely and that means that $a_{k} \rightarrow 0$ and for k 'large enough' is $\displaystyle |a_{k}|< \frac{1}{k^{\alpha}}$ where $\alpha>1$.
Unfortunately that is not true. There exist convergent series $\sum a_n$ of positive numbers, such that the condition $a_n<\frac1{n^\alpha}$ does not hold for all 'large enough' n, for any $\alpha>1$. For example, if $a_n = \dfrac1{n(\ln n)^2}$, then $\sum a_n$ converges (by the integral test), but $n^\alpha a_n\to\infty$ for all $\alpha>1$.

Originally Posted by hermanni
Let .Show that there exists a sequence such that .
There is a standard trick to construct this example. Let $t_n$ denote the tail of the series $\sum|a_k|$, in other words $t_n = \sum_{k=n}^\infty|a_k|$. Then $t_n\to0$ as $n\to\infty$. Let $b_k = 1/\sqrt{t_k}$. Clearly $b_k\to\infty$ as $k\to\infty$.

For the clever proof that $\sum|a_kb_k|$ converges, see here.