Cauchy product of series

Prove that $\displaystyle \sum_{n=0}^{\infty } \frac{1}{n!} \cdot \sum_{n=0}^{\infty } \frac{(-1)^{n}}{n!}=1$.

Prove that $\displaystyle \left(\sum_{n=0}^{\infty } q^{n}\right)^{2}=\sum_{n=0}^{\infty } (n+1)q^{n} \quad (\left|q\right|<1)$.

Prove that the square of the convergent series $\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{n+1}}{\sqrt{n}}$ is a divergent series.

Prove that the product of the two series $\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{n-1}}{n^{\alpha }} \ (\alpha >0)$ and $\displaystyle \sum_{n=1}^{\infty } \frac{(-1)^{n-1}}{n^{\beta }} \ (\beta >0)$ is a convergent series when $\alpha +\beta >1$, and a divergent series when $\alpha +\beta <1$.

Verify that the product of the two divergent series $1-\displaystyle \sum_{n=1}^{\infty } \left(\frac{3}{2}\right)^{n}$ and $1+\displaystyle \sum_{n=1}^{\infty } \left(\frac{3}{2}\right)^{n-1}\left(2^{n}+\frac{1}{2^{n+1}}\right)$ is an absolutely convergent series.

Let $\sum_{n = 1}^{\infty } c_{n}$ be the series obtained by multiplying the series $\sum_{n = 1}^{\infty } a_{n}$ and $\sum_{n = 1}^{\infty } b_{n}$. Find $c_{n}$, if:

Show that the square of the convergent series $\sum_{n = 1}^{\infty } \frac{(-1)^{n}}{\sqrt{n}}$ is a divergent series.

Let $\alpha > 0, \beta > 0$. Show that the product of two convergent series

is a series that converges if $\alpha +\beta > 1$, and diverges if $\alpha +\beta < 1$.

Show that:

Show that the series obtained by multiplying two divergent series

converges absolutely.

Let the series $\sum_{n = 1}^{\infty } a_{n}$, where $a_{n} > 0(n \in N)$, converge, and let the series $\sum_{n = 1}^{\infty } b_{n}$, where $b_{n} > 0(n \in N)$, diverge. Show that the product of these series is a divergent series.

Prove that if the series

converge and their sums equal $A, B$, and $C$, respectively, then the equality $C = A B$ holds.

Prove that if the series $\sum_{n = 0}^{\infty } a_{n}$ and $\sum_{n = 0}^{\infty } b_{n}$ are summable by the Poisson-Abel method to the sums $A$ and $B$, respectively, then the series $\sum_{n = 0}^{\infty }\left(a_{0} b_{n}+a_{1} b_{n-1}+\ldots +a_{n-1} b_{n}+a_{n} b_{0}\right)$, i.e., the product of the given series, is also summable by the Poisson-Abel method to the sum $A B$.

What conditions must the series $u_{0} + u_{1} + u_{2} + \cdots + u_{n} + \cdots$ satisfy in order that its Cauchy product [Problem 34 of Part I, Problem 23 of Part II, Part VIII Chapter 1 5] with any convergent series $v_{0} + v_{1} + v_{2} + \cdots + v_{n} + \cdots$ results in a convergent series

Prove that the Cauchy product of two absolutely convergent series converges absolutely.