Definition of exponential function -

Question

A lot of textbooks offer a definition of the exponential function such as this:

$$\exp(x):=\sum_{k=0}^{\infty}\frac{x^k}{k!}.$$

a) Show that the given definition for $\exp$ is correct, meaning, show that the series is convergent for every $x\in \mathbb{R}$.

b) Show, with help of a geometric series, that

$$\exp(x)=\sum_{k=0}^N\frac{x^k}{k!}+R_{N+1}(x),$$

whereas

$$R_{N+1}(x)|\leq \frac{2|x|^{N+1}}{(N+1)!} \text{ for } x\in \mathbb{R} \text{ with } |x|\leq 1+\frac{N}{2}.$$

c) Show that $\exp$ solve the differential equation $\dot{x}=x$ on $\mathbb{R}$ and $\exp(0)=1$.

We started talking about functions consisting of sums (mostly series') and this is the first exercise in my textbook concerning that topic. Also, I never saw a definition of the exponential function such as this. The only thing that came to mind was that I thought about that the definition shown in b) reminds me of the taylor polynom, but I guess that doesn't have anything to with this one. To be honest I'm all lost here. I don't even know how to approach.

I would appreciate any help. I don't know how to solve this type of questions yet.

Edit:

b) $...\leq \frac{|x|^{N+1}}{(N+1)!}(\sum_{k=0}^{\infty}(\frac{x}{N+1})^k)$, right?

But applying the geometric series to it it shows that it diverges since $x$ can be greater than $1$ too, right?

And one silly question: The exercise says that I have to show

$\exp(x)=\sum_{k=0}^{N}\frac{x^k}{k!}+R_{N+1}(x)$,

I'm still not sure why one would have to do the estimates to prove that. To be honest I don't even know what there is to prove.

Answer 1

Hints:

1. For a power-series $\sum_{k=0}^\infty a_k x^k$ we can apply the ratio test to determine the convergence radius $R$. For your case $a_k = \frac{1}{k!}$ so $$\frac{1}{R} = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| = \lim_{n\to\infty}\frac{n!}{(n+1)!} = \ldots$$

2. Try to estimate the remainder. We have (using the triangle inequality) $$|R_{N+1}(x)| = \left|\sum_{k=N+1}^\infty \frac{x^k}{k!}\right| \leq \frac{|x|^{N+1}}{(N+1)!}\left(1 + |x|\frac{1}{N+2} + |x|^2\frac{1}{(N+2)(N+3)} + \ldots\right) \leq \frac{|x|^{N+1}}{(N+1)!}\left(1 + |x|\frac{1}{N+2} + |x|^2\frac{1}{(N+2)^2} + \ldots\right) = \ldots$$ The last series is a geometrical series. Sum it and apply $|x| < 1 + N/2$. You can read more here: Taylor series with the remainder term.

3. Differentiate $\frac{d}{dx}\sum_{k=0}^\infty \frac{x^k}{k!}$ term-by-term and finally shift the summation index to show that $\frac{d}{dx}\exp(x) = \exp(x)$. Plugging in $x=0$ gives $\exp(0) = 1$. These two facts is enough to prove the statement. See this question for more information.

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${\bf \text{More hints for b}}:$ The equation $\exp(x) = \sum_{k=0}^N\frac{x^k}{k!} + R_{N+1}(x)$ is the definiton of $R_{N+1}(x)$. The exercise is to show that this quantity satisfy the inequality given in the problem.

"But applying the geometric series to it it shows that it diverges since $x$ can be greater than 1 too, right?" Note that the geometrical series $\sum \left(\frac{|x|}{N+2}\right)^k$ converges for $\frac{|x|}{N+2} < 1$ i.e. $|x| < N+2$.