Question

I'm confused about what this question is asking me to do: In exercise 17, use the Lagrange form of the remainder to prove that the Maclaurin series converges to the generating function from the given exercise (#7).
#7 just gives you xe^x, so I don't understand what I am supposed to find the error with

Answer 1

find the general error for the n+1th term i spose. can you provide a pic of the question?

Answer 2

what is the maclaruin series for x e^x?

Answer 3

x^n+1/n!

Answer 4

\[e^x=\sum_{n=0}^{\infty}\frac{1}{n!}x^n\]
\[xe^x=\sum_{n=0}^{\infty}\frac{1}{n!}x^{n+1}\]

Answer 5

how do we determine the formula for truncation error with this now?

Answer 6

Substitute n+1 in for n

Answer 7

right, and?

Answer 8

Take the derivative at the maximum value on the interval?

Answer 9

Except for the fact that I was not provided an interval!

Answer 10

well, its valid for all x in R isnt it?

Answer 11

Yes, but you have to take the derivative at c, which is the highest point on the interval

Answer 12

well, for a given interval [a,b] the highest point is b

Answer 13

No, its whatever makes the nth derivative largest

Answer 14

well, prolly not accurate, but its a monotonic function for x=0 and above ...

Answer 15

f = x e^x

f' = x' e^x + x e^x = (1+x) e^x

f'' = (1+x)' e^x + (1+x) e^x = (2+x) e^x

f''' = (3+x) e^x

f^(n) = (n+x)e^x

right?

Answer 16

how do we determine what makes the nth derivative 'largest' with this?

Answer 17

I guess it is infinity

Answer 18

e^x dominates the linear factor .... im guess e^x is max as x to infinity?

then we need to prolly limit if c to infinity .. but thats just a guess.

Answer 19

I have absolutely no idea. My book is absolutely terrible. I had no idea what the question was asking in the first place

Answer 20

seems to be something like this

\[\lim_{c\to\infty}R_n=\lim_{c\to\infty}\left|\frac{(c+n)e^c~c^{n+1}}{(n+1)!}\right|\]

Answer 21

not =

but <=

Answer 22

We never learned anything like that

Answer 23

neither have i, but then i havent taken an error estimation course or a course with it in it.

Answer 24

This is calc 2

Answer 25

Or BC calc, whichever you prefer

Answer 26

https://www.youtube.com/watch?v=cMc6l8AhF9M

Answer 27

I know what Lagrange Error is; my confusion is how I am supposed to do it without an interval. I have no idea what they are referring to when they say the Maclaurin series or the generating function. They only gave me one equation

Answer 28

the link i sent explains it, im watching it now .. its a 10-15min video

Answer 29

I finished the video

Answer 30

we hav a Mac series, so its centered at 0

\[xe^x=\sum_0^{k}\frac{x^{n+1}}{n!}+R_n\]
\[f^{(n)}[xe^x]=(n+x)e^x\implies R_n\le\frac{(n+1+c)e^cx^{n+2}}{(n+1)!}\]
\[\lim_{n\to \infty} \frac{(n+1+c)e^c x^{n+2}}{(n+1)!}\]
\[e^c~\lim_{n\to \infty} \frac{(n+1+c)x^{n+2}}{(n+1)!}\]
if the limit goes to 0, then the Remainder converges
assuming my f^(n)(c) is correct

Answer 31

Oooooh, I see. So I guess that they want me to prove that xe^x converges to the series that I wrote for xe^x?

Answer 32

prove that the limit of the Remainder goes to 0 as n approaches infinity yes

Answer 33

OOOH, I GET WHAT THE QUESTION IS ASKING NOW!

Answer 34

Thanks so much for helping me!!!

Answer 35

youre welcome, sorry it took so long, but i had to basically learn it along the way :)

Answer 36

That was a horribly worded question. I really appreciate it