The tag has no wiki summary.

learn more… | top users | synonyms

62
votes
6answers
3k views

Roots of truncations of e^x - 1

During a talk I was at today, the speaker mentioned that if you truncate the Taylor series for $e^x - 1$, you'll get lots of roots with nonzero real part, even though the full Taylor series only has ...
17
votes
2answers
1k views

Integral representation of higher order derivatives

I'm quite curious about the following phenomena, that still puzzle me although I have a proof, and I'd be really glad if someone may shred some light, showing an interpretation or a generalization. I ...
6
votes
2answers
3k views

power series of the reciprocal… does a recursive formula exist for the coefficients. [closed]

Hello If $f(x)=\sum _{n=0}^{\infty } b_nx^n$, and $\frac{1}{f(x)}=\sum _{n=0}^{\infty } d_nx^n$. Then the coefficients of the reprical of f can be written down. The first few terms are: $d_0 = ...
3
votes
1answer
137 views

Approximate the square root of (1-X) efficiently through (nested) products

Currently, I encountered a problem of approximating the following series: $$ (I-X)^{-\frac{1}{2}}=I+\frac{1}{2}X+\frac{1\cdot3}{2\cdot4}X^{2}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6}X^{3}+\ldots $$ where ...