An example: a convergence series, a divergent series, whose Cauchy product is convergent.

Question

How to find an example: a convergence series $\sum a_n$, a divergent series $\sum b_n$, whose Cauchy product $\sum c_n$ with $c_n=\sum_{i+j=n}a_ib_j$ is convergent? Is there a simple example?

Answer 1

Let be $a_n$ such that $a_{2k}=a_{2k+1}$ for every $k=0,1,2...$. Let be $b_k=(-1)^k$. Then it is $$ c_{2n} = \sum\limits_{k = 0}^{2n} {a_k } b_{2n - k} = a_0 - a_1 + \cdots - a_{2n - 1} + a_{2n} $$ and $$ c_{2n + 1} = \sum\limits_{k = 0}^{2n + 1} {a_k } b_{2n - k} = - a_0 + a_1 + \cdots - a_{2n} + a_{2n + 1} $$ It follows that $$ s_{2n} = \sum\limits_{k = 0}^{2n} {c_k = a_0 } + a_2 + \cdots a_{2n} $$ and $$ s_{2n + 1} = \sum\limits_{k = 0}^{2n + 1} {c_k = a_1 } + a_3 + \cdots a_{2n + 1} $$ If the series $ \sum\limits_{k = 0}^{ + \infty } {a_k } $ is convergent we get the required example. For instance we can choose $a_0=a_1=1$, $a_2=a_3=1/2$, $a_4=a_5=1/2^2$ and so on.

Answer 2

HINT.- Take a limit of the form $0\cdot\infty$ and develop the corresponding Taylor series.