Show that \({I_0} = \frac{1}{2}(1 + {{\text{e}}^{ - \pi }})\) .

By letting \(y = x - n\pi \) , show that \({I_n} = {{\text{e}}^{ - n\pi }}{I_0}\) .

Hence determine the exact value of \(\int_0^\infty  {{{\text{e}}^{ - x}}|\sin x|{\text{d}}x} \) .

Consider the infinite geometric series

\[1 - {x^2} + {x^4} - {x^6} +  \ldots \;\;\;\left| x \right| < 1.\]

Show that the sum of the series is \(\frac{1}{{1 + {x^2}}}\).

Hence show that an expansion of \(\arctan x\) is \(\arctan x = x - \frac{{{x^3}}}{3} + \frac{{{x^5}}}{5} - \frac{{{x^7}}}{7} +  \ldots \)

\(f\) is a continuous function defined on \([a,{\text{ }}b]\) and differentiable on \(]a,{\text{ }}b[\) with \(f'(x) > 0\) on \(]a,{\text{ }}b[\).

Use the mean value theorem to prove that for any \(x,{\text{ }}y \in [a,{\text{ }}b]\), if \(y > x\) then \(f(y) > f(x)\).

(i)     Given \(g(x) = x - \arctan x\), prove that \(g'(x) > 0\), for \(x > 0\).

(ii)     Use the result from part (c) to prove that \(\arctan x < x\), for \(x > 0\).

Use the result from part (c) to prove that \(\arctan x > x - \frac{{{x^3}}}{3}\), for \(x > 0\).

Hence show that \(\frac{{16}}{{3\sqrt 3 }} < \pi  < \frac{6}{{\sqrt 3 }}\).

Prove by induction that \(n! > {3^n}\), for \(n \ge 7,{\text{ }}n \in \mathbb{Z}\).

Hence use the comparison test to prove that the series \(\sum\limits_{r = 1}^\infty  {\frac{{{2^r}}}{{r!}}} \) converges.

Show that \(n! \geqslant {2^{n - 1}}\), for \(n \geqslant 1\).

Hence use the comparison test to determine whether the series \(\sum\limits_{n = 1}^\infty  {\frac{1}{{n!}}} \) converges or diverges.