Python 麦克劳林系列ln(x+1)

Question

I have to write a program of Maclaurin series ln(x+1) on Python.

I need to use input function for two values: x, n. Then check if the values are legal and calculates the Maclaurin approximation (of order n) of the expression ln (1 + ) around the point x.

*Maclaurin series ln(x+1)= sum of ((-1)^n/n)*x^n

I stacked in the end when I calculate to expression, that what I wrote (after all the checks before):

for i in range(n + 1):
    if i <= 1:
        continue
    else:
        x = x + (((-1) ** (i + 1)) * (x ** i) / i)

When I input the test I get a number but it's a wrong answer.

Please help me understand what is wrong in this code.

Answer 1

You are modifying the value of x in each iteration of the loop. Add and then store the partial sums in another variable.

def maclaurin_ln(x, n):
    mac_sum = 0
    for i in range(1, n + 1):
        mac_sum += (((-1) ** (i + 1)) * (x ** i) / i)
    return mac_sum

You can test this with the built-in function log1p to see how close they can get.

1. For ln(2) for different n,

from tabulate import tabulate
res = []
for n in [1, 10, 100, 1000, 10000]:
    p = math.log1p(1)
    q = maclaurin_ln(1, n)
    res.append([1, n, p, q, q-p])
tabulate(res, headers=["x", "n", "log1p", "maclaurin_ln", "maclaurin_ln-log1p"])

  x      n     log1p    maclaurin_ln    maclaurin_ln-log1p
---  -----  --------  --------------  --------------------
  1      1  0.693147        1                  0.306853
  1     10  0.693147        0.645635          -0.0475123
  1    100  0.693147        0.688172          -0.004975
  1   1000  0.693147        0.692647          -0.00049975
  1  10000  0.693147        0.693097          -4.99975e-05

2. For different x,

res = []
for x in range(10):
    p = math.log1p(x/10)
    q = maclaurin_ln(x/10, 100)
    res.append([x/10, 1000, p, q, q-p])
tabulate(res, headers=["x", "n", "log1p", "maclaurin_ln", "maclaurin_ln-log1p"])

  x     n      log1p    maclaurin_ln    maclaurin_ln-log1p
---  ----  ---------  --------------  --------------------
0    1000  0               0                   0
0.1  1000  0.0953102       0.0953102           1.38778e-17
0.2  1000  0.182322        0.182322            2.77556e-17
0.3  1000  0.262364        0.262364           -1.11022e-16
0.4  1000  0.336472        0.336472            0
0.5  1000  0.405465        0.405465           -1.11022e-16
0.6  1000  0.470004        0.470004            5.55112e-17
0.7  1000  0.530628        0.530628           -4.44089e-16
0.8  1000  0.587787        0.587787           -9.00613e-13
0.9  1000  0.641854        0.641854           -1.25155e-07

Answer 2

Mathematically, the Maclaurin series is a bit beyond me, but I'll try to help. Two things.

First, you're storing all the successive values in x, as you calculate them; that means that the term for n = 5 (i = 5) is using a value of x which isn't the original value of the parameter x, but which has the successive results of the four previous computations stored in it. What you need to do instead is something like:

    total = 0
    for each value:
          this term = some function of x    # the value of x does not change
          total = total + this term

Second, why aren't you interested in the term when i (or n) is equal to 1? The condition

   if i <= 1:
       continue

skips out the case when i equals 1, which evaluates to -x.

That should fix it, as far as I can see.