Question

find the first three nontrivial terms in the power series expansions centered at zero for the following functions:
a) f(x) = e^(-2x) sin(x)

b) g(x) = x^2 3^x + sin(x+2)

Answer 1

someone help me please

Answer 2

\(\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-...\)

Answer 3

\[e^{-2x}=1-2 x+2 x^2-\frac{4 x^3}{3}+...\]

Answer 4

multiply the first few terms and see what you get

Answer 5

So you would get = x + 2x^5/3! + 2x^7/5! etc.....

How did you get the series for sin and e?

Answer 6

i memorized the one for sine
it is a very common function, good to just know it
here is an easy way to remember it:
\(\sin(0)=0\) so there is no constant term
and sine is odd, so only odd powers
and it alternates

Answer 7

also memorize the one for \(e^x\) which is very easy
\[\sum_{n=0}^{\infty}\frac{x^n}{n!}\]
then replace \(x\) by \(-2x\)

Answer 8

but i am fairly sure you did not multiply correctly

Answer 9

you have \(1\times x\) as one term
next lowest power comes from \(-2x\times x=-2x^2\)
then the cube term comes from \(-\frac{x^3}{3!}\times 1\)

Answer 10

so i think it begins
\[x-2x^2-\frac{x^3}{3}+...\]