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!
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.
The Maclaurin series for $e^{x}$ is 1 + x + $\frac{1}{2!}x^{2}$ + $\frac{1}{3!}x^{3}$ + $\frac{1}{4!}x^{4}$ + ... = $\sum_{n=0}^{\infty}$ $\frac{1}{n!}x^{n}$.
The Maclaurin series for $\sqrt{1+x}$ is 1 + $\frac{1}{2}x$ - $\frac{1}{8}x^{2}$ + $\frac{1}{16}x^{3}$ - $\frac{5}{128}x^{4}$ + ... = $\sum_{n=0}^{\infty}$ $\frac{(-1)^{n}(2n)!}{(1-2n)(n!)^{2}4^{n}}x^{n}$.
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 $e^{4x} \sqrt{1+x}$.
The first 5 nonzero terms of the Maclaurin series for $e^{4x} \sqrt{1+x}$ are 1 + 2$x$ + $\frac{9}{4}x^{2}$ + $\frac{5}{3}x^{3}$ + $\frac{17}{16}x^{4}$ + ...