Find First 5 Nonzero Terms of Maclaurin Series for $e^{4x} \sqrt{1+x}$

In summary, to find the first 5 nonzero terms in the Maclaurin series for $f(x)=e^{4x} \sqrt{1+x}$, one needs to first find a series for $e^{4x}$ and a series for $\sqrt{1+x}$ and then multiply them together. This concept may be difficult to understand as it is not explained properly in the book and the instructor did not provide clear explanations either. However, it is important to understand as there may be a similar question on the upcoming test.
  • #1
ineedhelpnow
651
0
find the first 5 nonzero terms in maclaurin series. (might be binomial)

$f(x)=e^{4x} \sqrt{1+x}$my book doesn't explain it properly and my instructor didnt explain it and I am very stuck and there's going to be one similar to this on the test. help!
 
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  • #2
ineedhelpnow said:
find the first 5 nonzero terms in maclaurin series. (might be binomial)

$f(x)=e^{4x} \sqrt{1+x}$my book doesn't explain it properly and my instructor didnt explain it and I am very stuck and there's going to be one similar to this on the test. help!

Find a series for $\displaystyle \begin{align*} \mathrm{e}^{4x} \end{align*}$, find a series for $\displaystyle \begin{align*} \left( 1 + x \right) ^{\frac{1}{2}} \end{align*}$ and then multiply them together...
 
  • #3
ok thanks. nice "seeing" you again Prove It. :)
 

FAQ: Find First 5 Nonzero Terms of Maclaurin Series for $e^{4x} \sqrt{1+x}$

What is a Maclaurin series?

A Maclaurin series is a type of power series expansion that represents a function as an infinite sum of powers of the variable, with each term being multiplied by a coefficient. It is named after Scottish mathematician Colin Maclaurin.

What is the Maclaurin series for $e^{x}$?

The Maclaurin series for $e^{x}$ is 1 + x + $\frac{1}{2!}x^{2}$ + $\frac{1}{3!}x^{3}$ + $\frac{1}{4!}x^{4}$ + ... = $\sum_{n=0}^{\infty}$ $\frac{1}{n!}x^{n}$.

What is the Maclaurin series for $\sqrt{1+x}$?

The Maclaurin series for $\sqrt{1+x}$ is 1 + $\frac{1}{2}x$ - $\frac{1}{8}x^{2}$ + $\frac{1}{16}x^{3}$ - $\frac{5}{128}x^{4}$ + ... = $\sum_{n=0}^{\infty}$ $\frac{(-1)^{n}(2n)!}{(1-2n)(n!)^{2}4^{n}}x^{n}$.

How do you find the Maclaurin series for $e^{4x} \sqrt{1+x}$?

To find the Maclaurin series for $e^{4x} \sqrt{1+x}$, we can use the formula for the product of two power series, which states that (a + b)(c + d) = ac + (ad + bc)x + bd$x^{2}$ + ..., where a, b, c, and d are constants. We can plug in the Maclaurin series for $e^{4x}$ and $\sqrt{1+x}$ and simplify to find the Maclaurin series for $e^{4x} \sqrt{1+x}$.

What are the first 5 nonzero terms of the Maclaurin series for $e^{4x} \sqrt{1+x}$?

The first 5 nonzero terms of the Maclaurin series for $e^{4x} \sqrt{1+x}$ are 1 + 2$x$ + $\frac{9}{4}x^{2}$ + $\frac{5}{3}x^{3}$ + $\frac{17}{16}x^{4}$ + ...

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