Taylor Series

Consider the taylor series expansion of $f(x)$ about the point $a$ ,

which is $\displaystyle \lim_{x \to a} f(x)=f(a)+\dfrac{f'(a)}{1!}(x-a)+\dfrac{f''(a)}{2!}(x-a)^2+\cdots$

Now , I want to ask can we write this as $\displaystyle \lim_{x \to a}f(x)=f(a)+\dfrac{f'(a)}{1!}(x-a)+\dfrac{f''(c)}{2!}(x-a)^2$ ?

That is, we are compressing all the infinite terms into a single term, where the derivative is evaluated about any other point $c$ in the domain of $f$ .

In this case, for example I modified the third term, but it could be done for any finite term in the series.

Can we prove this that for any particular $c$ , we can simply write the summation into above form?

#Calculus

Easy Math Editor

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Comments

There are many forms of the exact remainder for a Taylor’s series. The one closest to what you want is this....

For a sufficiently differentiable function, and for any $a,x$ and $n$ , there exists $c$ between $a$ and $x$ such that $f(x)=\sum_{j=0}^{n-1}\frac{f^{(j)}(a)}{j!}(x-a)^j +\frac{f^{(n)}(c)}{n!}(x-a)^n$

Mark Hennings - 1 year, 5 months ago

Yes sir, this is what I wanted...but I wanted to know the proof of the claim or if you could write a proof of it...or you could provide a reference from where I can read more about it.

Vilakshan Gupta - 1 year, 5 months ago

Put $F(t)=\sum_{j=0}^{n-1}\frac{f^{(j)}(t)}{j!}(x-t)^j \hspace{2cm} G(t)=(x-t)^n$ Then $F(a)=\sum_{j=0}^{n-1}\frac{f^{(j)}(a)}{j!}(x-a)^j \hspace{1cm} F(x)=f(x)\hspace{1cm}F’(t)=\frac{f^{(n)}(t)}{(n-1)!}(x-t)^{n-1}$ While $G(a)=(x-a)^n\hspace{1cm}G(x)=0\hspace{1cm}G’(t)=-n(x-t)^{n-1}$ Now use the Generalised Mean Value Theorem, to find $c$ between $a$ and $x$ such that $(F(x)-F(a))G’(c)=(G(x)-G(a))F’(c)$ Most textbooks on real analysis should give proofs of Taylor’s Theorem with remainder...

Mark Hennings - 1 year, 5 months ago

@Mark Hennings, @Chew-Seong Cheong, @David Vreken , @Chris Lewis

Vilakshan Gupta - 1 year, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$