The Mystery of the e Series: Uncovering Its Name

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The series in question, 1/1! + 2/2! + 3/3!... converges to e, but it does not have a specific name. It represents the Taylor series for e^x evaluated at x = 1, also known as the Maclaurin series. There was some confusion regarding the equivalence of this series and the series for e, but they are indeed both equal to e. The discussion also touches on the efficiency of calculating e using series versus the limit of (1 + 1/n)^n, noting that series converge more rapidly. Overall, the conversation highlights the mathematical properties and relationships of these series.
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I just wonder what's the name of the serries 1/1!+2/2!+3/3!... I know it equals e but I just whant to know how it's called.

PS. Titels of good books abot serries would allso be welcome.
 
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\displaystyle e=\sum \limits_{n=0}^{\infty} \frac{1}{n!}, if I remember correctly.
What are you interested in learning about series?
Most series, as far as I know, don't have names.
 
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Exercise: prove the series written in the first two posts are the same!


I don't think this particular series has a name. However, it is the evaluation of the Taylor series for e^x at x = 1, or more specifically, the MacLauren series.
 
Hurkyl, they are not the same. The series in the first post was
\frac{1}{1!}+ \frac{2}{2!}+ \frac{3}{3!}+...= 1+ 1+ \frac{1}{2!}+ ...
and so is e+ 1, not e.
 
They're both e.

e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}

\displaystyle e=\sum \limits_{n=0}^{\infty} \frac{1}{n!}=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}\ldots=1+1+\frac{1}{2!}+\ldots
 
They really are both e. I tried to prove thath and I think I maneged to prove that they are equal for very large n, where 1/n is almost equal to 0. It's actualy qouit easy to do it with the use of the binomical expresion.

PS. But I still don't get it why it's so much fester to do it with a series. When you get to the 13'th element (13/13!) it's allready excet to 10 digits. But if you do it as (1+1/n) on n, you have to use a very large n to get such an excet figure. Why is that?

PPS. Thanks for the info Hurkley.
 
What? I'm wrong? Moi?? Oh, blast, I started my series with n=1 instead of n= 0!
 
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