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Sudharaka

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**mbeaumont99's question from Math Help Forum,**

This is to do with some infinite summation work that we are going through at college at the moment. We have the function \(\displaystyle t_{n} =\frac{(x\ln a)^{n}}{n!}\) and have been substituting in different x and a values to determine a general statement for the infinite summation of the function. I have found that \(\displaystyle S_{n}=a^{x}\)

I need to do a formal proof for this general statement and heard that a Taylor series would be able to do that. If anyone would be able to start me off on this, send me in a different direction, or simply contribute to this thread then I would be extremely appreciative.

Thanks,

mbeaumont99

Hi mbeaumont99,

One thing you can do is to find the Taylor series expansion of \(f(x)=a^{x}\) and see whether it is \(\displaystyle \sum t_{n}\). The Taylor series for the function \(f \) around a neighborhood \(b\) is,

\[f(x)=\sum_{n=0}^{\infty}\frac {f^{(n)}(b)}{n!} \, (x-b)^{n}\]

Of course I am assuming here that the function \(f\) can be expressed as a Taylor series expansion around a neighborhood of \(b\) (that is \(f\) is analytic). To get a more detailed idea about what functions are analytic read this and this. We shall use \(b=0\) so that we get the Maclaurin's series.

\[f(x)=\sum_{n=0}^{\infty}\frac {f^{(n)}(0)}{n!} \, x^{n}\]

Now we have to find out \(f^{(n)}(0)\) with regard to the function \(f(x)=a^{x}\). Differentiating \(f\) a couple of times we can "feel" that \(f^{n}(x)=a^x(\ln(a))^n\,\forall\,n\in\mathbb{N}=\mathbb{Z}\cup\{0\}\). To prove this in a formal manner we shall use mathematical induction.

When \(n=0\), the result is obvious. We shall assume that the result is true for \(n=p\in\mathbb{N}\). That is,

\[f^{(p)}(x)=a^{x}(\ln(a))^p\]

Now consider, \(f^{(p+1)}(x)\).

\[f^{(p+1)}(x)=\frac{d}{dx}f^{(p)}(x)=(\ln(a))^p \frac{d}{dx}a^x=a^x(\ln(a))^{p+1}\]

Therefore by Mathematical induction, \(f^{n}(x)=a^x(\ln(a))^n\,\forall\,\in\mathbb{N}\)

\[\therefore f^{n}(0)=(\ln(a))^n\,\forall\,\in\mathbb{N}\]

Hence,

\[f(x)=\sum_{n=0}^{\infty}\frac{(\ln(a))^n}{n!}\, x^{n}=\sum_{n=0}^{\infty}t_{n}\]