1
vote
1answer
52 views

Rate of convergence of an exponential function

If I have a function $$f = \exp(\sqrt{n} \cdot \frac{\sqrt{\log{n}}}{\sqrt{n}-\sqrt{\log n}}),$$ I can notice, that $$\lim_{n \to \infty} f = \infty,$$ but also I can notice that it goes very slowly ...
2
votes
5answers
241 views

Does the series $\,\displaystyle\sum_{n = 1}^{\infty}\left(2^{1/n} - 1\right)\,$ converge?

I'm trying to determine if the following sum converges or diverges (this is question 38 in section 11.7 of Stewart's Early Transcendentals): $$\sum_{n = 1}^{\infty}(2^{1/n} - 1)$$ I've considered ...
3
votes
1answer
61 views

Proof of $ e^x-x^2 \gt 1 $ when $ x \gt 0$ and $x$ is a real number .

I want to Prove $ e^x-x^2 \gt 1 $ when $ x \gt 0$ and $x$ is a real number . For this purpose , my trying is as the following : $ e^x-x^2 = \{1+x+\dfrac{x^2}{2!} + +\dfrac{x^3}{3!}++\dfrac{x^4}{4!}+ ...
0
votes
1answer
49 views

Convergence rate of exponential function

If I have two exponential function, say $f_1(t)=4e^{-3t}+6e^{-7t}$ and $f_2(t)=\frac{2e^{-3t}+5e^{-7t}}{e^{-3t}+9e^{-7t}} - 2$ who are all converge to $0$. Then, the convergence rate of $f_1(t)$ can ...
1
vote
2answers
42 views

Which Expression of e Converges Fastest?

Writing a program, would be helpful to know whether expressing e as the sum of the reciprocals of the factorials (as in the Taylor series expansion of $e^x$ with $x=1$) converges quicker than ...
3
votes
3answers
78 views

Convergence of $\sum _{k=1}^\infty (1-\frac{1}{k})^{k^2}$

Found the alternative form: $\sum _{k=1}^\infty ((1-\frac{1}{k})^{k})^k$. Tried various criteria, no luck so far.
3
votes
0answers
54 views

How general is the convergence of the exponential function's power series?

Let $\mathbf{V}$ be a Fréchet space whose underlying set is $V$. Let $\;\; \beta \: : \: V\times V \: \to \: V \;\;$ be a continuous bilinear map that has an identity element and is ...
2
votes
2answers
106 views

What is $ \lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$?

How to solve the following limit question? $$\lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$$ Thanks a lot.
1
vote
1answer
125 views

Exponential as power series

Is there a function that does not depend on $a$ such that $\sum_{x=1}^\infty \frac{a^x}{x!}f(x) = \mathrm e^{-a}$? Just to be clear, the summation starting from 1 is intentional, otherwise the ...
1
vote
1answer
1k views

Radius of convergence for the exponential function

I'm studying physics and am currently following a course on complex analysis and in the section on analytic functions, the radius of convergence $R$ for power series was introduced. The Taylor ...
1
vote
2answers
192 views

Find the limit of $\frac{\bar{z}}{z}$ as $z$ goes to $0$.

I put it in exponential form to get $\dfrac{re^{-i \theta}}{re^{i \theta}}$ but I think I'll get $\frac{0}{0}$ which isn't defined and isn't a good enough proof to say it doesn't have a limit.
2
votes
3answers
128 views

Find an exponential function with given condition

How can I have an example of an exponential function defined in the X range 1 - infinity, with values starting at 40 and converging to 1?