Binomial Theorem & Power Series (Taylor & Maclaurin)
Binomial Theorem
1. Simplify the following:
a) $\frac{(n+1)!}{(n-1)!} - n!$ b) $\frac{(n+1)!}{(n-1)!}$
2. Find the appropriate value(s) for $n$ satisfying $(n+1)! + n! = 72 (n-1)!$
3. Expand $(p - 2q)^4$ using Pascal's triangle
4. Expand $\left(x - \frac{1}{x}\right)^6$ using Binomial's Theorem
5. Find the coefficient of $x^{15}$ in the expansion of $\left(x^3 - \frac{1}{x^2}\right)^9$
6. Determine $(2 + x)^5$
a) From expansion above, find $(2 - x)^5$ b) Hence, evaluate $(2.1)^5 - (1.9)^5$ correct to two decimal places
7. Find the first four terms in the expansion of the following functions in ascending powers of $x$ and state the range of convergence values of $x$ for which the expansion is valid:
a) $\frac{1}{\sqrt{1+x}}$
b) $(8 - x)^{\frac{1}{3}}$
8. Let the following radical functions be given:
$$\sqrt{1 - \frac{x}{4}}, \quad \sqrt{2 + x}, \quad \sqrt{1 + 4x}, \quad \sqrt[3]{4 - \frac{x}{2}}$$ Suggest the most suitable function from the list to approximate $\sqrt{2}$ and justify
9. Expand $\sqrt{1 - x}$ as a series in ascending powers of $x$ up to the term $x^3$ and state the range of convergence values of $x$. Hence, approximate $\sqrt{0.98}$, giving to five decimal places. Compare your answer with calculator's.
10. Let the function be
$$f(x) = \frac{x+7}{x^2 - x - 6}$$ a) Express $f(x)$ in its partial fraction form b) Determine the first four terms of expansion for $f(x)$ and state the range of convergence of values of $x$
Power Series: Taylor & Maclaurin Series
1. Find the first four nonzero terms of Taylor series for the given function below expanded about the given value of $a$:
a) $f(x) = \cos x ; ; ; a = \frac{\pi}{6}$
b) $f(x) = \frac{1}{1-x} ; ; ; a = 0$
c) $f(x) = \ln(3 + x) ; ; ; a = 1$
2. Find the first three nonzero terms of Taylor series for $f(x) = x^{\frac{2}{3}}$ about $a = 1$. Write in the notation of $\sum$ or $C$.
Hence, find the approximation for $(1.03)^{\frac{2}{3}}$.
Source: FAD1014 TUTORIAL 11 25-26 PASUM.pdf