(1)

\begin{align*} \sum_{j=0}^{\infty}b_{j}x^{j} & =\sum_{j=0}^{\infty}\sum_{k=0}^{j}C\left(j,k\right)a_{k}x^{j}\\ & =\sum_{k=0}^{\infty}a_{k}\sum_{j=k}^{\infty}C\left(j,k\right)x^{j}\\ & =\sum_{k=0}^{\infty}a_{k}x^{k}\left(1-x\right)^{-\left(k+1\right)}\\ & =\frac{1}{1-x}\sum_{k=0}^{\infty}a_{k}\left(\frac{x}{1-x}\right)^{k} \end{align*}

(2)

\begin{align*} \sum_{j=0}^{\infty}b_{j}\frac{x^{j}}{j!} & =\sum_{j=0}^{\infty}\sum_{k=0}^{j}C\left(j,k\right)a_{k}\frac{x^{j}}{j!}\\ & =\sum_{k=0}^{\infty}a_{k}\sum_{j=k}^{\infty}C\left(j,k\right)\frac{x^{j}}{j!}\\ & =\sum_{k=0}^{\infty}a_{k}\frac{x^{k}}{k!}e^{x}\\ & =e^{x}\sum_{k=0}^{\infty}a_{k}\frac{x^{k}}{k!} \end{align*}

(3)

\begin{align*} \sum_{j=0}^{\infty}b_{j}x^{j} & =\sum_{j=0}^{\infty}\sum_{k=0}^{j}\left(-1\right)^{k}C\left(j,k\right)a_{k}x^{j}\\ & =\sum_{k=0}^{\infty}\left(-1\right)^{k}a_{k}\sum_{j=k}^{\infty}C\left(j,k\right)x^{j}\\ & =\sum_{k=0}^{\infty}\left(-1\right)^{k}a_{k}x^{k}\left(1-x\right)^{-\left(k+1\right)}\\ & =\frac{1}{1-x}\sum_{k=0}^{\infty}a_{k}\left(\frac{-x}{1-x}\right)^{k} \end{align*}

(4)

\begin{align*} \sum_{j=0}^{\infty}b_{j}\frac{x^{j}}{j!} & =\sum_{j=0}^{\infty}\sum_{k=0}^{j}\left(-1\right)^{k}C\left(j,k\right)a_{k}\frac{x^{j}}{j!}\\ & =\sum_{k=0}^{\infty}\left(-1\right)^{k}a_{k}\sum_{j=k}^{\infty}C\left(j,k\right)\frac{x^{j}}{j!}\\ & =\sum_{k=0}^{\infty}\left(-1\right)^{k}a_{k}\frac{x^{k}}{k!}e^{x}\\ & =e^{x}\sum_{k=0}^{\infty}a_{k}\frac{\left(-x\right)^{k}}{k!} \end{align*}