Question

Write the formula for\({b_8}\) for a given\(f(x) = \sum\limits_{n = 0}^\infty {{b_n}} {(x - 5)^n}\).

Short answer

The formula for \({b_8}\) is\({b_8} = \frac{{{f^{(8)}}(5)}}{{40,320}}\)

Step-by-step solution

Use theorem

If\(f\)has a power series representation at\(a\), which can be written as\(f(x) = \sum\limits_{n = 0}^\infty {{c_n}} {(x - a)^n}\quad |x - a| < R\),

Then, its coefficient are given by the formula\({c_n} = \frac{{{f^{(n)}}(a)}}{{n!}}\)

As given, the function is \(f(x) = \sum\limits_{n = 0}^\infty {{b_n}} {(x - 5)^n}\) for all \(x\)

Compare the given function with the general power series representation, it is noticed that \(a = 5\) and\({b_n} = \frac{{{f^{(n)}}(a)}}{{n!}}\)

Obtain\({b_8}\)

Substitute 8 for \(n\) and 5 for \(a\) in\({b_n}\)

\(\begin{aligned}{l}{b_8} &= \frac{{{f^{(8)}}(5)}}{{8!}}\\{b_8} &= \frac{{{f^{(8)}}(5)}}{{40,320}}\end{aligned}\)

Hence, the required formula for \({b_8}\) is\({b_8} = \frac{{{f^{(8)}}(5)}}{{40,320}}\).