2
votes
1answer
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.
6
votes
1answer
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 ...
1
vote
1answer
52 views

Why can we first take the limit that goes to e?

For example \begin{equation} \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}} ...
19
votes
3answers
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 ...
3
votes
0answers
59 views

Roots of derivative of q-expontial function

Let the q-deformation of the exponential function be defined by $$ e_q(z)=\sum_{n=0}^\infty{\frac{z^n}{[n]_q!}}. $$ Eq. (1.8) of this paper provides the product representation $$ ...
0
votes
1answer
83 views

Series proof for $e^x$.

Problem: Prove $$\sum_{n=0}^\infty \frac{1}{n!}x^n=e^x$$ I am a bit confused on how I should start this proof. Any pointers on how I should start would help.
0
votes
1answer
35 views

First order ODE with $f'(x) = 810(10)^x$

I'm trying to find an explicit form of the series $f(0) = 89.1,f(1) = 899.1,f(2) = 8999.1, \cdots$. My first though was to take the derivative and integrate it, which I've done before with a fair ...
8
votes
3answers
122 views

Showing $n!<e(\frac{n}{2})^n$

I'd like to prove that $n!<e(\frac{n}{2})^n$. What I have so far: $\sqrt[n]{n!} = \sqrt[n]{1\cdot 2 \cdot \ldots \cdot n} \leq \frac{1+\ldots +n}{n}=\frac{(n+1)n}{2n}=\frac{(n+1)}{2}$. Thus ...
2
votes
4answers
203 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 ...
1
vote
2answers
102 views

Show that $\sum\limits_{n=1}^\infty \frac{1}{n^z}$ converges absolutely

I want to show that $\large\sum\limits_{n=1}^\infty \frac{1}{n^z}$ converges absolutely for $\Re(z) > 1$, so I want to show that $\large\sum\limits_{n=1}^\infty |\frac{1}{n^z}|$ converges, or ...
0
votes
1answer
31 views

Multi part problem to prove functional relation of the exponential function

I'm worried about part (i) right now mostly. Part 3 is easy, and part 2 I can probably get after some work. I know that $\exp(-z) = \large\sum\limits_{n=0}^\infty \frac{-z^n}{n!} = ...
0
votes
3answers
135 views

How to find the sum of this power series $\sum\limits_{n=0}^\infty \frac {x^{5n}} {(5n)!}$

How to prove that $$ \sum\limits_{n=0}^\infty \frac {x^{5n}} {(5n)!}= \frac{2}{5} e^{-\cos \left( 1/5\,\pi \right) x}\cos \left( \sin \left( 1/5\,\pi \right) x \right) +\frac{2}{5}\, e^{\cos ...
3
votes
0answers
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 ...
3
votes
1answer
65 views

How to show that $e^{x+y} = e^x e^y$ by series expansion [duplicate]

I know that $e^xe^y=e^{x+y}$ but I want to show it by expanding the exponentials in MacLaurin Series. $$ \left(\sum_{n=0}^{\infty} \frac{x^n}{n!}\right) \left(\sum_{m=0}^{\infty} ...
1
vote
2answers
259 views

Derivative of exponential function proof

I'm looking for a straight forward proof using the definition of a derivative applied to the exponential function and substitution of one of the limit definitions of $e$, starting with $e = ...
2
votes
3answers
157 views

Prove that the limit definition of the exponential function implies its infinite series definition.

Here's the problem: Let $x$ be any real number. Show that $$ \lim_{m \to \infty} \left( 1 + \frac{x}{m} \right)^m = \sum_{n=0}^ \infty \frac{x^n}{n!} $$ I'm sure there are many ways of pulling this ...
6
votes
4answers
495 views

Evaluating the precision in the calculation of $\mathrm{e}$

I'm calculating $\mathrm{e}$ using a computer like this: $$ \mathrm{e} \approx \sum\limits_{i=0}^n {1\over i!} $$ I'm storing it as a rational number. I was wondering, if I write down my rational ...
5
votes
0answers
112 views

How prove this inequality $\frac{1}{n!}\sum\limits_{k=0}^{\infty}\frac{k^n}{k!}\ge e(C\ln{n})^{-n}$

Show that: $$\dfrac{e^n}{(\ln{n})^n}\ge \dfrac{1}{n!}\sum_{k=0}^{\infty}\dfrac{k^n}{k!}\ge e(C\ln{n})^{-n},\ n\ge 2$$ where $C>e$ is constant. My try: ...
1
vote
2answers
75 views

does this series converge / diverge

does this series converge/diverge conditionally or absolutly $\sum_{n=2}^{\infty} (-1)^n \cdot \frac{\sqrt{n}}{(-1)^n + \sqrt{n}} \cdot \sin(\frac{1}{\sqrt{n}})$ i can use the facts that: ...
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.
1
vote
1answer
15 views

finding sequence for e converging at some speed

I want to find an infinite sequence that conerges to e so that the kth term of the sequence is less than 10^-k away from e. Obviously, I've considered the Taylor series, but asymptotic bounds on the ...
7
votes
2answers
129 views

What are the properties of the roots of the incomplete/finite exponential series?

Playing around with the incomplete/finite exponential series $$f_N(x) := \sum_{k=0}^N \frac{z^k}{k!} \stackrel{N\to\infty}\longrightarrow e^z$$ for some values on alpha (e.g. ...
0
votes
0answers
50 views

splitting up summations of product

I want to check the validity of splitting up summations of products. I am using the DFT matrix and am trying to get a simplified expression of it . In essence, I am trying to prove the following lemma ...
0
votes
1answer
127 views

Matrix Exponential equality

I was reading about the matrix exponential function and I came across this: If $xy = yx$ then $$ \exp(x+y) = \exp(x)\cdot\exp(y) $$ My textbook gives a proof as follows: $$ \exp(x+y) = ...
6
votes
6answers
414 views

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 ...
3
votes
1answer
100 views

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 ...
3
votes
3answers
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.
0
votes
1answer
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 | ...
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
3answers
92 views

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
10
votes
2answers
775 views

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.
1
vote
2answers
113 views

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?
1
vote
1answer
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 ...
5
votes
3answers
98 views

Stuck on this integral involving exp and the floor function

Here is the integral $$\int_0^\infty \lfloor x \rfloor e^{-x}dx$$ Here is what I have so far: $$I = \sum_{n=0}^\infty \int_n^{n+1} n e^{-x}dx$$ $$ = \sum_{n=0}^\infty -ne^{-n-1} + ne^{-n}$$ $$ = ...
0
votes
1answer
621 views

Prove that the exponential function is sequentially continuous?

I am supposed to use the definition of the exponential function to prove that if x is a real number and the modulus of x is less than 1, the modulus of exp(x)-1 is less than or equal to (e-1)*modulus ...
1
vote
1answer
440 views

Use Cauchy's Multiplication Theorem and the Binomial Theorem to prove $\exp(x+y)=\exp(x)\exp(y)$

I am to use Cauchy's Multiplication Theorem and the Binomial Theorem in order to prove $\exp(x+y)=\exp(x)\exp(y) $ but I have no idea where to begin. All I can think of doing is setting $\exp(x)$ ...
2
votes
2answers
431 views

Coefficients of exponential generating functions

I have a problem in which I need to solve for a number of sequences using exponential generating functions. I understand how finding the coefficients of ordinary generating functions work, however, I ...
4
votes
1answer
153 views

Euler's Question

I came across this problem that I would like to ask you about: For which values $a>0$ does there $\exists$ a limit of the sequence $$a, a^{a},a^{a^{a}}, a^{a^{a^{a}}}...$$ Well this looks like a ...
1
vote
2answers
185 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
1answer
131 views

Finding $\pi$ through sine and cosine series expansions

I am working on a problem in Partha Mitra's book Observed Brain Dynamics (the problem was originally from Rudin's textbook Real and Complex Analysis, and appears on page 54 of Mitra's book). ...