Obtain the Maclaurin polynomials $p_1\left(x\right)$ and $p_2\left(x\right)$ 

${\ⅇ}^x$$\stackrel{\text{series in x}}{\to }$$1+x+\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+\frac{1}{24}{x}^4+\frac{1}{120}{x}^5+\frac{1}{720}{x}^6+\frac{1}{5040}{x}^7+\frac{1}{40320}{x}^8$$\stackrel{\text{assign to a name}}{\to }$$p_1$

$\ln \left(1\+x\right)$$\stackrel{\text{series in x}}{\to }$$x-\frac{1}{2}{x}^2+\frac{1}{3}{x}^3-\frac{1}{4}{x}^4+\frac{1}{5}{x}^5-\frac{1}{6}{x}^6+\frac{1}{7}{x}^7-\frac{1}{8}{x}^8$$\stackrel{\text{assign to a name}}{\to }$$p_2$

Form the (sorted) product $p_1\left(x\right)\cdot p_2\left(x\right)$ 

Although the expanded product can be obtained via the Context Panel, the sort option in the Context Panel does not provide for sorting with the powers ascending. Hence, the sort and expand commands have been applied directly.

$\mathrm{sort}\left(\mathrm{expand}\left(p_1\cdot p_2\right)\,x\,\mathrm{ascending}\right)$

$x+\frac{1}{2}{x}^2+\frac{1}{3}{x}^3+\frac{3}{40}{x}^5-\frac{7}{144}{x}^6+\frac{23}{504}{x}^7-\frac{29}{720}{x}^8-\frac{1291}{17280}{x}^9-\frac{53539}{1209600}{x}^{10}-\frac{9697}{604800}{x}^{11}-\frac{1457}{345600}{x}^{12}-\frac{527}{604800}{x}^{13}-\frac{253}{1693440}{x}^{14}-\frac{1}{47040}{x}^{15}-\frac{1}{322560}{x}^{16}$

Obtain the degree 11 Maclaurin polynomial for the product $f\left(x\right)\cdot g\left(x\right)$ 

Write the product $e^x \ln \left(1\+x\right)$, being sure to use the exponential "e" taken from one of the palettes, or from Command Completion.

Context Panel: Series≻Series≻$x$ (See Figure 8.5.17(a) for guidance completing the Series dialog.)

${\ⅇ}^x\cdot \ln \left(1\+x\right)$$\stackrel{\text{series in x}}{\to }$$x+\frac{1}{2}{x}^2+\frac{1}{3}{x}^3+\frac{3}{40}{x}^5-\frac{7}{144}{x}^6+\frac{23}{504}{x}^7-\frac{29}{720}{x}^8+\frac{629}{17280}{x}^9-\frac{120287}{3628800}{x}^{10}+\frac{607337}{19958400}{x}^{11}$

Form the Cauchy product of the coefficients of the series for $f\left(x\right)$ and $g\left(x\right)$ 

${\ⅇ}^x$$\stackrel{\text{formal series}}{\to }$$\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}$

$\ln \left(1\+x\right)$$\stackrel{\text{formal series}}{\to }$$\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\mathrm{-1}\right)}^n{x}^{n+1}}{n+1}$

Write $a_n$ and $b_n$ as Maple functions

$a\left(n\right)\=1\/n\!$$\stackrel{\text{assign as function}}{\to }$$a$

$b\left(n\right)\={\left(-1\right)}^n\/\left(n\+1\right)$$\stackrel{\text{assign as function}}{\to }$$b$

Form $c_n$, the coefficient of the Cauchy product, as a Maple function

$c\left(n\right)\=\sum _{k\=0}^{n}a\left(k\right)\cdot b\left(n-k\right)$$\stackrel{\text{assign as function}}{\to }$$c$

Display a sequence of coefficients $c_n$ 

Write $c\left(n\right)$

Context Panel: Sequence≻$n$ 
Complete the Sequence dialog as per Figure 8.5.17(b)ib

$c\left(n\right)$$\stackrel{\text{sequence w.r.t. n}}{\to }$$1,\mathrm{hypergeom}\left(\left[\mathrm{-1}\,1\,1\right]\,\left[2\right]\,1\right),\frac{\mathrm{hypergeom}\left(\left[\mathrm{-2}\,1\,1\right]\,\left[2\right]\,1\right)}{2},\frac{\mathrm{hypergeom}\left(\left[\mathrm{-3}\,1\,1\right]\,\left[2\right]\,1\right)}{6},\frac{\mathrm{hypergeom}\left(\left[\mathrm{-4}\,1\,1\right]\,\left[2\right]\,1\right)}{24},\frac{\mathrm{hypergeom}\left(\left[\mathrm{-5}\,1\,1\right]\,\left[2\right]\,1\right)}{120},\frac{\mathrm{hypergeom}\left(\left[\mathrm{-6}\,1\,1\right]\,\left[2\right]\,1\right)}{720},\frac{\mathrm{hypergeom}\left(\left[\mathrm{-7}\,1\,1\right]\,\left[2\right]\,1\right)}{5040},\frac{\mathrm{hypergeom}\left(\left[\mathrm{-8}\,1\,1\right]\,\left[2\right]\,1\right)}{40320},\frac{\mathrm{hypergeom}\left(\left[\mathrm{-9}\,1\,1\right]\,\left[2\right]\,1\right)}{362880},\frac{\mathrm{hypergeom}\left(\left[\mathrm{-10}\,1\,1\right]\,\left[2\right]\,1\right)}{3628800}$$\stackrel{\text{simplify}}{\=}$$1,\frac{1}{2},\frac{1}{3},0,\frac{3}{40},-\frac{7}{144},\frac{23}{504},-\frac{29}{720},\frac{629}{17280},-\frac{120287}{3628800},\frac{607337}{19958400}$