Step 1

Note that for even n the coefficient in the sum is zero, so some terms are absent.

Step 2

\(\displaystyle{S}_{{{10}}}{\left({x}\right)}={\frac{{{8}}}{{\pi}}}{\left({\sin{\ }}{x}\ +\ {\frac{{{1}}}{{{27}}}}\ {\sin{\ }}{3}{x}\ +\ {\frac{{{1}}}{{{125}}}}\ {\sin{\ }}{5}{x}\ +\ {\frac{{{1}}}{{{273}}}}\ {\sin{\ }}{7}{x}\ +\ {\frac{{{1}}}{{{729}}}}\ {\sin{\ }}{9}{x}\right)}\)

Step 3

We see that \(\displaystyle{S}_{{{10}}}{\left({x}\right)}\) is a very good a approximation of the function

\(\displaystyle{f{{\left({x}\right)}}}={x}{\left(\pi\ -\ {x}\right)}.\)

The absolute value of their difference is given below for the comparison:

Note that for even n the coefficient in the sum is zero, so some terms are absent.

Step 2

\(\displaystyle{S}_{{{10}}}{\left({x}\right)}={\frac{{{8}}}{{\pi}}}{\left({\sin{\ }}{x}\ +\ {\frac{{{1}}}{{{27}}}}\ {\sin{\ }}{3}{x}\ +\ {\frac{{{1}}}{{{125}}}}\ {\sin{\ }}{5}{x}\ +\ {\frac{{{1}}}{{{273}}}}\ {\sin{\ }}{7}{x}\ +\ {\frac{{{1}}}{{{729}}}}\ {\sin{\ }}{9}{x}\right)}\)

Step 3

We see that \(\displaystyle{S}_{{{10}}}{\left({x}\right)}\) is a very good a approximation of the function

\(\displaystyle{f{{\left({x}\right)}}}={x}{\left(\pi\ -\ {x}\right)}.\)

The absolute value of their difference is given below for the comparison: