Does anyone know what the fastest converging equation is for calculating e? I have found many different equations when I have been searching google, but I was wondering which ones can yield the most digits of e in the fewest terms.  Thank you.
122406
What are other faster methods of calculating $e^x$ up to any number of decimals other than using the Taylor series formula?
5019
What is the best approach to calculating the largest prime factor of a number?  I'm thinking the most efficient would be the following:   Find lowest prime number that divides cleanly Check if result of division is prime If not, find next lowest Go to 2.   I'm basing this assumption on it being easier to calculate the small prime factors. Is this about right? What other approaches should I look into?  Edit: I've now realised that my approach is futile if there are more than 2 prime factors in play, since step 2 fails when the result is a product of two other primes, therefore a recursive algorithm is needed.  Edit again: And now I've realised that this does still work, because the last found prime number has to be the highest one, therefore any further testing of the non-prime result from step 2 would result in a smaller prime.
81519
I just ran into an expression, and I would like to know what it converges to. $$\sum^{\infty}_{n=1} \frac{n}{(n-1)!}$$  Do you know if it converges to something (like $e$) or, in alternative, how to find out what it converges to?
92208
I have seen a post here that says that you can convert an exponential generating function into an ordinary one with the aid of the Laplace transform. Is it possible to do the reverse transformation? i.e. I want to convert an ordinary generating function into an exponential one.
103505
I was wondering if there are functions for which $$f'(x) &gt; f(x)$$ for all $x$. Only examples I could think of were $e^x - c$ and simply $- c$ in which $c &gt; 0$. Also, is there any significance in a function that is always less than its derivative?    Edit: Thank you very much for all the replies. It seems almost all functions that apply are exponential by nature...  Are there more examples like - 1/x?  Again are there any applications/physical manifestations of these functions? [for example an object with a velocity that is always greater than its position/acceleration  is always greater than its velocity]
95762
As a follow-up to a , I'd like to know if  $$\sum_{n=1}^\infty\frac{1}{2 ^ n}$$  is convergent? If yes, then what does it converge to?  Just as a sidenote - the whole issue started with reading about Zeno's Turtle Paradox which made me curious about the underlying mathematics.
67707
We use multiplication for repeated addition, and in turn use exponents for repeated multiplication. What topic comes after this, for repeated exponentials? Is there something my teachers are hiding from me?
53786
This question is just a soft question, about can we compute a general formula for everything? Or it has some restrictions?  Like $\int x^ndx=\frac{x^{n+1}}{n+1}+C$.  I am not able to deduce a formula for both.  I dont know where to start, so please guide me?
43287
i am pulling my hair out in solving this problem. i know, it is a stupid question but i am not that good at maths, and many thanks for any help   $\lim_{n \to \infty} \frac{2^{n+1}+3^{n+1}}{2^n+3^n}$  my steps are: $\lim_{n \to \infty} \frac{2^{n+1}+3^{n+1}}{2^n+3^n} = \frac{2^{n}2+3^{n}3}{2^n+3^n} = help = ?  $  i know, it diverges as the denominator grows faster than the nominator, but the calculation is killing me now
35500
I was wondering on the following and I probably know the answer already: NO. Is there another number with similar properties as $e$? So that the derivative of $\exp(x)$ is the same as the function itself. I can guess that it's probably not, because otherwise $e$ wouldn't be that special, but is there any proof of it?
81503
I know that  $A_{[a} B_{b]} = \frac{1}{2!}(A_{a}B_{b} - A_{b}B_{a})$  But how can write $E_{[a} F_{bc]}$ like the above?  Can you provide a reference where this notational matter is discussed?
97123
I'm very eager to know and understand the definition of $e$. Textbooks define $e$ as follows $$ e = \lim_{p\to\infty} \left[1+\frac1{p}\right]^p \approx 2.71828 $$ Is there an &quot;easy to understand&quot; proof of this? I'm really looking for a derivation of this which is very intuitive and easy to comprehend. By the way I'm watching .
60502
How do you prove that $e=\sum_{n=0}^{\infty}\frac{1}{n!}$? Here I am assuming $e:=\lim_{n\to\infty}(1+\frac{1}{n})^n$. Do you have any good PDF file or booklet available online on this? I do not like how my analysis text handles this...
83864
In fact the real question is: Does $\lim\limits_{n\to\infty}1^{n}=e$?.  I know that   $$ \lim\limits_{n\to\infty}\left(1+\dfrac{1}{n}\right)^n=e, $$  So, can we say that $1^\infty=e$?  And, by logic, This product  $$\underset{\text{infinity times}}{1\cdot1\cdot\ldots\cdot1},$$ gives $1$ not other value.
13240
Is there a direct way to determine how many digits a power of 2 will contain without actually performing the multiplication?  An estimation would help as well if there is no absolute solution.  EDIT: In both decimal and binary bases.
72099
So I have the following recurrence relation for the growth rate of an algorithm:  $T(n)$ = time taken by algorithm to solve problem of size n:  $$T(n) = T(n-1) + T(\lceil(n/2)\rceil)$$  Clearly this does not obey any polynomial law but now my question is what does it obey?  I merely want an asymptotic equivalence.  My guess is subexponential since this is faster than polynomial but clearly slower than any exponential form.  What though is the question....
74105
Calculus book:     Find all numbers $a$ such that $\forall x, a^x \ge 1+x$    I immediately thought of the inequality $e^x\ge 1+x$ and guessed that the answer was any number $a$ in $[e,\infty)$. After playing around with a graphing app I can see this is definitely true, although I can't explain why.   Why is $e$ specifically the smallest number such that the inequality holds? To my untrained eye, a quick scan of the equation does not give rise to anything involving the constant $e$. Maybe because $1+x$ is only a tangent to the equation $a^x$ when $a=e$? Thanks.