<p>Given any well-behaved function \(\mathrm{f}(x)\), using Maclaurin’s series expansion, it can be expressed as a Polynomial of \(x\):</p>
<p style="text-align: center">\(\mathrm{f}(x)={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+{{a}_{3}}{{x}^{3}}+....+{{a}_{n}}{{x}^{n}}\)</p>
<p>The series’ accuracy increases when:</p>
<ol class="num-list"><li><div class="list-item-paragraph">more terms are used.</div></li><li><div class="list-item-paragraph">when the value of \(x\) is close to zero.</div></li></ol>

A power series represents a function, on a suitable interval, as a polynomial with indefinitely many terms:

\[\mathrm{f}(x)=a_0+a_1x+a_2x^2+a_3x^3+\cdots+a_nx^n+\cdots\]

Within its interval of convergence, retaining more terms can improve the approximation near the expansion point. This is not a global guarantee outside that interval.

Power Series

Power Series