Tagged Questions

66 views

Summation of exponential series [duplicate]

Evaluate the limit: $$\lim_{n \to \infty}e^{-n}\sum_{k = 0}^n \frac{n^k}{k!}$$ It is not as easy as it seems and the answer is definitely not 1. Please help in solving it.
358 views

Approximating the exponential function

I have found experimentally something that seems graphically like an approximation of the exponential function. However, it is totally experimental and I have no idea whether it really converges ...
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Why can we first take the limit that goes to e?

For example \begin{aligned} \lim_{n \to \infty} \left(1 + \frac{1}{ \frac{n-1}{2}} \right)^{n} &= \lim_{n \to \infty} \left(1 + \frac{1}{ \frac{n-1}{2}} ...
347 views

How to prove $\lim_{n \to \infty} e^n \cdot \left( \sum_{k=0}^{n-1} ({k-n \over e})^k/k! \right)- 2 \cdot n = \frac 23$?

I observed for the function $$f(n)= e^n \sum_{k=0}^{n-1}\left(\dfrac{k - n}{e}\right)^k \cdot \dfrac{1}{k!} \tag 1$$ with small $n$ that ...
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59 views

Which series of numbers effectively translates the factorial to the exponential function?

We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum ...
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Why is $\sum_{n=0}^{\infty}\frac{x^n}{n!} = e^x$?

I am trying to see where this relationship comes from: $\displaystyle \sum_{n=0}^{\infty}\frac{x^n}{n!} = e^x$ Does anyone have any special knowledge that me and my summer math teacher doesn't know ...
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Harlan J. Brothers's approximation to $e$ ad infinitum?

Consider the series generated by Harlan J. Brothers's method for the number $e$ http://en.wikipedia.org/wiki/List_of_representations_of_e Can they be improved again and again or is there a limit so ...
394 views

Exponential function and uniform convergence of polynomials.

How can I prove that no sequence of polynomials converges uniformly to the exponential function? Thanks in advance for any help.
78 views

How to calculate a result based on an exponentially changing pattern/sequence. [duplicate]

Given any number (including decimals) between 12.5 and 87.5 I need to calculate another number based on these results:  Input = 12.5 | 31.25 | 50 | 68.75 | 87.5 Result = 12.5 | 64.5  | 78 | ...
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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.
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Infinite Series

How can you show that $$\left(1-\frac{2}{n^2}\right)^{n^2/2} \le \frac{1}{e}\:\: \qquad\forall n \ge 2$$ Any ideas? Infinite series have never really been my thing. Thanks
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Show that $e^{A+B}=e^A e^B$

If $A$ and $B$ are $n\times n$ matrices such that $AB = BA$ (that is, $A$ and $B$ commute), show that $$e^{A+B}=e^A e^B$$ Note that $A$ and $B$ do NOT have to be diagonalizable.
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Exponential equality

How do I can prove that $$e\ :=\ \lim_{n\rightarrow\infty}\left(1 + \frac{1}{n}\right)^n\ =\ \lim_{n\rightarrow\infty}\sum_{j=1}^{n}\frac{1}{j!},$$ without use of derivatives?
306 views

Sum the series : $\sum_{n\geq 1} \frac{C(n,0)+C(n,1)+C(n,2)+\cdots+C(n,n)}{P(n,n)}$

Sum the series $$\sum_{n\geq 1} \frac{C(n,0)+C(n,1)+C(n,2)+\cdots+C(n,n)}{P(n,n)}$$ The answer is given as $e^2-1$. For getting that answer, $C(n,0)+C(n,1)+C(n,2)+\cdots+C(n,n)$ should be equal ...
98 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.