# Quantum Chemistry

Chapter04 - Home
Exercises: 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 - 14 - 15 - 16 - 17 - 18 - 19 - 20 - 21 - 22 - 23 - 24 - 25 - 26 - 27 - 28 - 29 - 30 - 31 - 32 - 33 - 34 - 35 - 36 - 37 - 38 - 39 - 40 - 41 - 42 - 43 - 44 - 45 - 46 - 47 - 48 - 49 - 50

Exercise. This is the solution to exercise 4.1 in the book.

Solution. If we differentiate $n$ times we get:

${f}^{\left(n\right)}\left(x\right)={c}_{n}\cdot n!+\sum _{k=n+1}^{\infty }{c}_{k}{\left(x-a\right)}^{k}$

Hence:

${f}^{\left(n\right)}\left(a\right)={c}_{n}\cdot n!$

or

${c}_{n}=\frac{{f}^{\left(n\right)}\left(a\right)}{n!}$

Thus, when applied to each coefficient we end up with:

$f\left(x\right)=\sum _{n=0}^{\infty }\frac{{f}^{\left(n\right)}\left(a\right)}{n!}{\left(x-a\right)}^{n}$