# Taylor series

#### Jameson

Staff member
I just realized that ILS gave a solution using a definition of a Taylor Series I hadn't seen before. The logic and result are clearly correct but I thought about trying it from the definition I am used to.

This might be wrong so I am hoping someone can check it and comment if necessary. The end result is correct but there is one spot where I am unsure, so read with scrutiny and skepticism.
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We are given $$\displaystyle \frac{s^n}{n!}\frac{d^n}{dq^n}(e^{-q^2})$$ and want to try to match it to this form: $$\displaystyle \sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n!} (x-a)^n$$

There is one term that they share which represents the nth derivative at $a$. These must match up so $f(a)=e^{-q^2}$. There are two things we need to know however, $f(x)$ and $a$. There are perhaps various combinations to try but the one that comes to mind first is $f(x)=e^{-x^2}$ and $a=q$.

Looking at $s^n$ it seems that this should match up with $(x-a)^n$ so we find that $x-a=s$ and since $a=q$ that is equivalent to $x-q=s$. Finally that gives us that $x=s+q$. If we plug that into $f(x)=e^{-x^2}$ then we have our result.

#### Klaas van Aarsen

##### MHB Seeker
Staff member
I just realized that ILS gave a solution using a definition of a Taylor Series I hadn't seen before. The logic and result are clearly correct but I thought about trying it from the definition I am used to.

This might be wrong so I am hoping someone can check it and comment if necessary. The end result is correct but there is one spot where I am unsure, so read with scrutiny and skepticism.
---------------------------------
We are given $$\displaystyle \frac{s^n}{n!}\frac{d^n}{dq^n}(e^{-q^2})$$ and want to try to match it to this form: $$\displaystyle \sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n!} (x-a)^n$$

There is one term that they share which represents the nth derivative at $a$. These must match up so $f(a)=e^{-q^2}$. There are two things we need to know however, $f(x)$ and $a$. There are perhaps various combinations to try but the one that comes to mind first is $f(x)=e^{-x^2}$ and $a=q$.

Looking at $s^n$ it seems that this should match up with $(x-a)^n$ so we find that $x-a=s$ and since $a=q$ that is equivalent to $x-q=s$. Finally that gives us that $x=s+q$. If we plug that into $f(x)=e^{-x^2}$ then we have our result.
Looks good.
I wasn't sure about $f(a)=e^{-q^2}$, although it is logical and fits. Pretty much anything that involves making conclusions about functions in this way makes me nervous. Not being able to rigorously prove a claim I'm using makes me feel like I don't have a right to use the claim, even if it's correct.