Use the binomial series to find the Maclaurian series for the function .
A)
B)
C)
D)
E)

Question

Use the binomial series to find the Maclaurian series for the function  . ​
A) ​ 
B) ​ 
C) ​ 
D) ​ 
E) ​

Quizplus · Accepted Answer

The binomial series is $(1+x)^\alpha = 1+\alpha x + \frac{\alpha(\alpha-1)}{2!}x^2+...$, where $\alpha$ is a constant. We can rewrite the given function as $11e^{\frac{1}{2}\ln\left(\frac{1-x^2}{1+x^2}\right)}$. Letting $\alpha=\frac{1}{2}$ and $x^2=-t$, we can write the function as $11\left(\frac{1+t}{1-t}\right)^{\frac{1}{2}}$. We can now apply the binomial series to get the Maclaurin series $ 11 + \frac{11}{2}t +\frac{11}{2}\cdot\frac{1}{2}\cdot\frac{-1}{2!}t^2 + \frac{11}{2}\cdot\frac{1}{2}\cdot\frac{-1}{2!}\cdot\frac{3}{4}\cdot\frac{-1}{3!}t^3+...$. Substituting back for $t$ and simplifying gives the answer $11eac44f_ec4c_86f6_bedf_336a0df6434a_TB7497_11$

Use the Binomial Series to Find the Maclaurian Series for the Function