To solve the problem, we start with the given equation:
\[
\log_e\left(\frac{1+x}{1-x}\right) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots
\]
### Step 1: Use the properties of logarithms
We can rewrite the left-hand side using the property of logarithms that states \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\):
\[
\log_e(1+x) - \log_e(1-x)
\]
### Step 2: Expand the logarithms using Taylor series
The Taylor series expansion for \(\log(1+x)\) around \(x=0\) is:
\[
\log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots
\]
Similarly, for \(\log(1-x)\):
\[
\log(1-x) = -\left(x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \ldots\right)
\]
### Step 3: Substitute the expansions into the equation
Substituting the expansions into our equation gives:
\[
\log(1+x) - \log(1-x) = \left(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots\right) - \left(-x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - \ldots\right)
\]
### Step 4: Simplify the expression
Combining the terms results in:
\[
\log(1+x) - \log(1-x) = x + x + \left(\frac{x^3}{3} + \frac{x^3}{3}\right) + \ldots
\]
This simplifies to:
\[
2x + \frac{2x^3}{3} + \frac{2x^5}{5} + \ldots
\]
### Step 5: Identify the coefficients
From the simplified expression, we can identify the coefficients:
- \(a_0 = 0\)
- \(a_1 = 2\)
- \(a_3 = \frac{2}{3}\)
- \(a_5 = \frac{2}{5}\)
### Step 6: Analyze the coefficients
Now we need to determine the relationship between \(a_1\), \(a_3\), and \(a_5\):
- \(a_1 = 2\)
- \(a_3 = \frac{2}{3}\)
- \(a_5 = \frac{2}{5}\)
### Step 7: Check if they are in a progression
To check if \(a_1\), \(a_3\), and \(a_5\) are in harmonic progression, we can use the property of harmonic progression:
Three numbers \(a\), \(b\), and \(c\) are in harmonic progression if:
\[
\frac{1}{a}, \frac{1}{b}, \frac{1}{c} \text{ are in arithmetic progression.}
\]
Calculating the reciprocals:
- \(\frac{1}{a_1} = \frac{1}{2}\)
- \(\frac{1}{a_3} = \frac{3}{2}\)
- \(\frac{1}{a_5} = \frac{5}{2}\)
Now, checking the arithmetic progression:
\[
\frac{3}{2} - \frac{1}{2} = 1 \quad \text{and} \quad \frac{5}{2} - \frac{3}{2} = 1
\]
Since the differences are equal, \(a_1\), \(a_3\), and \(a_5\) are indeed in harmonic progression.
### Final Answer
Thus, \(a_1\), \(a_3\), and \(a_5\) are in harmonic progression.
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