5.19 Maclaurin series Quiz

1. Which is the Maclaurin series for $\cos x$ up to the term in $x^{4}$?

• A. $1-\frac{x^{2}}{2}+\frac{x^{4}}{4}$
• B. $1-\frac{x^{2}}{2}+\frac{x^{4}}{24}$
• C. $x-\frac{x^{3}}{6}$
• D. $1+\frac{x^{2}}{2}+\frac{x^{4}}{24}$

2. Find the first three non-zero terms of the Maclaurin series for $e^{-x^{2}}$.

• A. $1-x^{2}+\frac{x^{4}}{2}$
• B. $1-x^{2}+\frac{x^{4}}{24}$
• C. $1+x^{2}+\frac{x^{4}}{2}$
• D. $1-x^{2}-\frac{x^{4}}{2}$

3. Find the first three terms of the series for $(1-x)^{-1}$.

• A. $1-x+x^{2}$
• B. $1+x-x^{2}$
• C. $1+x+x^{2}$
• D. $1-x-x^{2}$

4. Find the coefficient of $x^{2}$ in the Maclaurin series for $e^{x}\ln(1+x)$.

• A. $-\frac12$
• B. $1$
• C. $\frac32$
• D. $\frac12$

5. Given $\frac{1}{1+x^{2}}=1-x^{2}+x^{4}-\dots$, integrating term by term gives the series for:

• A. $\ln(1+x^{2})=x-\frac{x^{3}}{3}+\dots$
• B. $\arctan x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\dots$
• C. $\arcsin x=x-\frac{x^{3}}{3}+\dots$
• D. $\sin x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\dots$