Question

Use any method to find the taylor series for the function f(x)=xe^(x)-x centered at a=0. Write it in summation form.

@agent0smith

Answer 1

http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries_files/eq0025M.gif
(it's at the very top of the pic)

Answer 2

okay i know that formula now what?:)

Answer 3

oh im sorry i know what to do for this one i wrote the wrong problem!!

Answer 4

Not 100% sure how to write it in summation form, but theres a couple of examples here http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx

Answer 5

lets still work it out just in case i am wrong...
you take the first, second, and third derivative. then solve for f(0) and the derivatives. then plug those into the formula and by the formula you can create the summation form

Answer 6

summation form is just like for example e^x=1+x^2/2!+x^3/3!+....

Answer 7

you could do that, but it would be best to commit
\[e^x=\sum\frac{x^n}{n!}\] to memory

Answer 8

making
\[xe^x=\sum\frac{x^{n+1}}{n!}\]

Answer 9

its actually n+1 in the numerator

Answer 10

and memory doesnt help work the problem out for an exam that requires showing work but knowing it off hand would help btu the steps are critical to know

Answer 11

yeah i know, but it did say "using any method" right? so if you know one you can adjust to find the others, without starting from scratch

Answer 12

\(e^x, \sin(x), \cos(x), \ln(1+x)\) are ones that would be good just to know

Answer 13

yeah i have those memorized and a few others

Answer 14

theres always partial credit! lol

Answer 15

yeah but i am pretty sure (i could be wrong) that you can use your knowledge of one to find a similar one without starting from the beginning
for example if you know \(e^x=\sum\frac{x^n}{n!}\) (which is one definition) then you know \(xe^x\) as well

Answer 16

similarly if you know \(\sin(x)\) then you can find \(\frac{\sin(x)}{x}\) or \(x\sin(x)\) by just multiplying or dividing

Answer 17

yeah thats true but u still have to show all the steps unfortunately

Answer 18

k, but the line "Use any method " suggests that you need not start from first principles

Answer 19

btw as an aside, if you want the mclauren series for \(\frac{\sin(x)}{x}\) you pretty much have to divide, because the function is not defined at \(x=0\)

Answer 20

okay thank you

Answer 21

so i am willing to bet that a decent answer for full marks for this one is to say
\[e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...\\
xe^x=x+x^2+\frac{x^3}{2!}+\frac{x^4}{3!}+...\\
xe^x-x=x^2+\frac{x^3}{2!}+\frac{x^4}{3!}+...\]