Question 1

Using the formula

$\sum_{1}^{\infty} a_{n}x^{n}= f(0)+\frac{f^{1}(0)}{1!}x+\frac{f^{2}(0)}{2!}x^{2}+\frac{f^{3}(0)}{3!}x^{3}...$

Find the Maclaurin series of $f(x)=\sin{x}$

Accepted Answer

$x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}...$

Answer

$1-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}...$

Answer

$1-\frac{x^{2}}{3!}+\frac{x^{4}}{5!}-\frac{x^{6}}{7!}...$

Answer

$x-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}...$

Question 2

Find the Maclaurin series of $\cos{x}$ using the fact that $\frac{d}{dx}sin{x}=\cos{x}$ if you so wish

Accepted Answer

$1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}...$

Answer

$x-\frac{x^{3}}{3!}+\frac{x^{ 5}}{5!}-\frac{x^{7}}{7!}...$

Answer

$x-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}...$

Answer

$1-\frac{x^{3}}{3!}+\frac{x^{ 5}}{5!}-\frac{x^{7}}{7!}...$

Question 3

Using the formula

$\sum_{1}^{\infty} a_{n}x^{n}= f(0)+\frac{f^{1}(0)}{1!}x+\frac{f^{2}(0)}{2!}x^{2}+\frac{f^{3}(0)}{3!}x^{3}...$

Find the Maclaurin series of $f(x)=	an{x}$

Accepted Answer

$x+\frac{x^{3}}{3}+\frac{2x^{5}}{15}...$

Answer

$x+\frac{x^{2}}{2}+\frac{3x^{5}}{10}...$

Answer

$1+\frac{x^{3}}{3}+\frac{2x^{5}}{15}...$

Answer

$1+\frac{x^{2}}{2}+\frac{3x^{5}}{10}...$

Question 4

Find the Maclaurin series of $\sec^{2}{x}$ using the fact that:

$\frac{d}{dx}	an{x}=\sec^{2}{x}$

Accepted Answer

$1+x^{2}+\frac{2}{3}x^{4}...$

Answer

$x+x^{3}+\frac{2}{3}x^{5}...$

Answer

$1+x^{3}+\frac{2}{3}x^{5}...$

Answer

$x+x^{2}+\frac{2}{3}x^{4}...$

Question 5

Find the Maclaurin series of $2\sin{x}$

Accepted Answer

$2x-\frac{x^{3}}{3}+\frac{x^{5}}{60}...$

Answer

$2-\frac{x^{2}}{3}+\frac{x^{4 }}{60}...$

Answer

$2x-\frac{x^{2}}{3}+\frac{x^{4}}{60}...$

Answer

$2-\frac{x^{3}}{3}+\frac{x^{5}}{60}...$

Question 6

Find the Maclaurin Series of $e^{x^{2}}$

Accepted Answer

$1+x^{2}+\frac{x^{4}}{2}+\frac{x^{6}}{6}...$

Answer

$x+2x^{2}+\frac{2x^{4}}{2}+\frac{2x^{6}}{6}...$

Answer

$2x+2x^{2}+\frac{2x^{4}}{2}+\frac{2x^{6}}{6}...$

Answer

$x+x^{2}+\frac{x^{4}}{2}+\frac{x^{6}}{6}...$

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