Let $\sum_{n=0}^{\infty} a_n$, $\sum_{n=0}^{\infty} b_n$, and $\sum_{n=0}^{\infty} c_n$ be three series, referred to as the “a-series”, “b-series”, and “c-series” in the following. Let 
\[
c_n = \sum_{k=0}^{n} a_k b_{n-k}.
\]
\begin{enumerate}
  
    \item[(b)] Assume that the a-series and b-series both converge conditionally. Show that it is possible for the c-series to converge.
    
\end{enumerate}

We are asked to show that the series $ \sum_{n=0}^{\infty} c_n $ can converge, given that the series $ \sum_{n=0}^{\infty} a_n $ and $ \sum_{n=0}^{\infty} b_n $ both converge conditionally. This involves understanding the relationship between the two series, the convolution formula, and the convergence properties of infinite series.

### The Series $ c_n $
The given relation for $ c_n $ is a convolution of the sequences $ \{a_n\} $ and $ \{b_n\} $, defined as:

\[
c_n = \sum_{k=0}^{n} a_k b_{n-k}.
\]

This convolution describes the coefficients $ c_n $ as a sum of products of terms from the sequences $ a_n $ and $ b_n $.

### Conditional Convergence of $ a_n $ and $ b_n $
When a series converges conditionally, it means that the series converges, but the absolute value of the terms does not:

\[
\sum_{n=0}^{\infty} a_n \text{ converges conditionally} \quad \text{and} \quad \sum_{n=0}^{\infty} b_n \text{ converges conditionally}.
\]

That is, both series $\sum a_n$ and $\sum b_n$ converge, but $\sum |a_n|$ and $\sum |b_n|$ diverge.

### Key Step: The Convolution of Two Conditionally Converging Series

Now we examine whether the series for $ \sum_{n=0}^{\infty} c_n $ can converge, given that it is generated by the convolution of the two conditionally converging series. In particular, we consider the sum of $ c_n $:

\[
\sum_{n=0}^{\infty} c_n = \sum_{n=0}^{\infty} \left( \sum_{k=0}^{n} a_k b_{n-k} \right).
\]

By rearranging the double sum, we recognize this expression as:

\[
\sum_{n=0}^{\infty} c_n = \sum_{k=0}^{\infty} a_k \sum_{n=k}^{\infty} b_{n-k}.
\]

### Possible Convergence of the Series $ \sum_{n=0}^{\infty} c_n $

- The key property to investigate here is whether the convolution of two conditionally converging series can lead to a convergent series. A famous result in this context is *Toeplitz’s theorem*, which states that the convolution of two conditionally converging series can converge, even though each individual series does not converge absolutely.

- Specifically, Toeplitz's theorem assures us that **if two series converge conditionally**, their convolution can still converge, even if the individual series are not absolutely convergent. This provides the possibility that $ \sum_{n=0}^{\infty} c_n $ could converge.

### Conclusion

Thus, while the series $ \sum_{n=0}^{\infty} a_n $ and $ \sum_{n=0}^{\infty} b_n $ converge conditionally, it is possible for the series $ \sum_{n=0}^{\infty} c_n $ (the c-series) to converge, according to the properties of convolution and Toeplitz's theorem.

Would you like further clarification or more details on any part of the proof? 

Here are five related questions:
1. What is the difference between conditional and absolute convergence?
2. How does Toeplitz’s theorem apply to the convolution of series?
3. Can two divergent series have a convergent convolution?
4. What happens to the convolution of two absolutely convergent series?
5. Can you give an example of conditionally converging series whose convolution converges?

**Tip:** Rearranging sums in a double series requires careful attention to convergence properties, especially for conditionally converging series.

Let $\sum_{n=0}^{\infty} a_n$, $\sum_{n=0}^{\infty} b_n$, and $\sum_{n=0}^{\infty} c_n$ be three series, referred to as the 'a-series', 'b-series', and 'c-series' in the following. Let 
\[ c_n = \sum_{k=0}^{n} a_k b_{n-k}. \]
(b) Assume that the a-series and b-series both converge conditionally. Show that it is possible for the c-series to converge.

Explore the conditional convergence of series and how the convolution of two conditionally converging series can still converge, according to Toeplitz's theorem. This detailed solution involves infinite series, the convolution formula, and key convergence properties.

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