Create logarithmic_series.py (#12344)

* Create logarithmic_series.py

This is an implementation of logarithmic series in Python.
Reference: https://math.stackexchange.com/questions/3973429/what-is-a-logarithmic-series

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* Update logarithmic_series.py

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* Update logarithmic_series.py

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* updating DIRECTORY.md

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <cclauss@me.com>
Co-authored-by: cclauss <cclauss@users.noreply.github.com>

Author: 3 people

DIRECTORY.md

@@ -835,6 +835,7 @@
     * [Harmonic](maths/series/harmonic.py)
     * [Harmonic Series](maths/series/harmonic_series.py)
     * [Hexagonal Numbers](maths/series/hexagonal_numbers.py)
+    * [Logarithmic Series](maths/series/logarithmic_series.py)
     * [P Series](maths/series/p_series.py)
   * [Sieve Of Eratosthenes](maths/sieve_of_eratosthenes.py)
   * [Sigmoid](maths/sigmoid.py)

maths/series/logarithmic_series.py

@@ -0,0 +1,54 @@
+"""
+This is an implementation of logarithmic series in Python.
+Reference: https://math.stackexchange.com/questions/3973429/what-is-a-logarithmic-series
+"""
+
+
+def logarithmic_series(x_value: float, n_terms: int = 5, expand: bool = False) -> list:
+    """
+    Returns the logarithmic series for a number x (log x) upto n terms.
+
+    Parameters:
+        x_value: a floating point number for log(x)
+        n_terms: number of terms to be computed
+        expand: Set this flag to get the terms as real numbers,
+                unset for unsolved expressions
+
+    Examples:
+        >>> logarithmic_series(3)
+        ['(2^1)/1', '-(2^2)/2', '(2^3)/3', '-(2^4)/4', '(2^5)/5']
+
+        >>> logarithmic_series(-3)
+        ['-(4^1)/1', '(4^2)/2', '-(4^3)/3', '(4^4)/4', '-(4^5)/5']
+
+        >>> logarithmic_series(3, 6)
+        ['(2^1)/1', '-(2^2)/2', '(2^3)/3', '-(2^4)/4', '(2^5)/5', '-(2^6)/6']
+
+        >>> logarithmic_series(3, expand=True)
+        [2.0, -2.0, 2.6666666666666665, -4.0, 6.4]
+    """
+    n_times_x_minus_1: float = x_value - 1
+    n: int = 1
+    series: list = []
+    for _ in range(n_terms):
+        if expand:
+            series.append(((-1) ** (n + 1)) * (n_times_x_minus_1 / n))
+            n_times_x_minus_1 *= x_value - 1
+        else:
+            sign: str = "-" if (-1) ** (n + 1) == -1 else ""
+            term: str = (
+                sign + "(" + str(x_value - 1) + "^" + str(n) + ")" + "/" + str(n)
+            )
+            if term.startswith("-(-"):
+                term = "(" + term[3::]
+            elif term.startswith("(-"):
+                term = "-(" + term[2::]
+            series.append(term)
+        n += 1
+    return series
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()