The code below is run in Python 3.6 (Anaconda install), sympy 1.1.1:
#!/usr/bin/env python
import sympy as spy
import sympy.abc as abc
from sympy import pprint
n = abc.n
N = abc.N
EY = 2*N*(1-(1-1/(2*N))**n)
pprint(n-EY)
print('-'*80, '\n')
pprint((n-EY).series(N, x0=spy.oo).simplify())
print('-'*80, '\n')
pprint((n-EY).series(N, x0=spy.oo))
print('-'*80, '\n')
pprint((n-EY).series(N, x0=spy.oo).removeO().simplify())
When the code is run, I see the following output:
⎛ n ⎞
⎜ ⎛ 1 ⎞ ⎟
- 2⋅N⋅⎜- ⎜1 - ───⎟ + 1⎟ + n
⎝ ⎝ 2⋅N⎠ ⎠
--------------------------------------------------------------------------------
⎛1 ⎞
O⎜──; N → ∞⎟
⎜ 6 ⎟
⎝N ⎠
--------------------------------------------------------------------------------
6 5 4 3 2 5 4 3 2
n n 17⋅n 5⋅n 137⋅n n n n 7⋅n 5⋅n
n
───── - ──── + ───── - ──── + ────── - ─── - ──── + ─── - ──── + ──── -
──
23040 1536 4608 512 11520 192 1920 192 384 192
80
────────────────────────────────────────── +
─────────────────────────────── +
5 4
N N
4 3 2 3 2 2
n n 11⋅n n n n n n n
─── - ── + ───── - ── - ── + ── - ── ── - ─
192 32 192 32 24 8 12 4 4 ⎛1 ⎞
───────────────────── + ────────────── + ────── + O⎜──; N → ∞⎟
3 2 N ⎜ 6 ⎟
N N ⎝N ⎠
--------------------------------------------------------------------------------
⎛ 4 3 ⎛ 2 ⎞ 2 ⎛ 3 2
⎞
n⋅⎝5760⋅N ⋅(n - 1) + 960⋅N ⋅⎝- n + 3⋅n - 2⎠ + 120⋅N ⋅⎝n - 6⋅n + 11⋅n -
6⎠ +
──────────────────────────────────────────────────────────────────────────────
⎛ 4 3 2 ⎞ 5 4 3 2
12⋅N⋅⎝- n + 10⋅n - 35⋅n + 50⋅n - 24⎠ + n - 15⋅n + 85⋅n - 225⋅n +
274⋅n
──────────────────────────────────────────────────────────────────────────────
5
23040⋅N
⎞
- 120⎠
───────
simplify() seems to be removing all terms in the asymptotic expansion
(second printed item). The third printed item is correct and does not use
simplify(). The fourth print item, which uses simplify() after removing the
O(), is also correct.
Question: If I want to factor()/simplify() each term in the asymptotic
expansion, what is the best way to do that?
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