Resolve `(1)/(x^(6)(x+1))` into partial fractions.

Answer

To resolve the expression \(\frac{1}{x^6(x+1)}\) into partial fractions, we will follow these steps:

### Step 1: Set up the partial fraction decomposition
We start by expressing \(\frac{1}{x^6(x+1)}\) in terms of its partial fractions. The general form for the decomposition will be:

\[ \frac{1}{x^6(x+1)} = \frac{a}{x+1} + \frac{b}{x} + \frac{c}{x^2} + \frac{d}{x^3} + \frac{e}{x^4} + \frac{f}{x^5} + \frac{g}{x^6} \]

### Step 2: Combine the fractions
Next, we will combine the right-hand side over a common denominator:

\[ \frac{1}{x^6(x+1)} = \frac{a x^6 + b x^5(x+1) + c x^4(x+1) + d x^3(x+1) + e x^2(x+1) + f x(x+1) + g(x+1)}{x^6(x+1)} \]

### Step 3: Simplify the numerator
Now, we will simplify the numerator:

\[ 1 = a x^6 + b x^6 + b x^5 + c x^5 + c x^4 + d x^4 + d x^3 + e x^3 + e x^2 + f x^2 + f x + g x + g \]

Combining like terms, we get:

\[ 1 = (a + b)x^6 + (b + c)x^5 + (c + d)x^4 + (d + e)x^3 + (e + f)x^2 + (f + g)x + g \]

### Step 4: Set up equations by comparing coefficients
Now we will compare coefficients from both sides of the equation:

1. Coefficient of \(x^6\): \(a + b = 0\)
2. Coefficient of \(x^5\): \(b + c = 0\)
3. Coefficient of \(x^4\): \(c + d = 0\)
4. Coefficient of \(x^3\): \(d + e = 0\)
5. Coefficient of \(x^2\): \(e + f = 0\)
6. Coefficient of \(x\): \(f + g = 0\)
7. Constant term: \(g = 1\)

### Step 5: Solve the equations
From \(g = 1\), we can substitute back to find the other coefficients:

1. From \(f + g = 0\): \(f + 1 = 0 \Rightarrow f = -1\)
2. From \(e + f = 0\): \(e - 1 = 0 \Rightarrow e = 1\)
3. From \(d + e = 0\): \(d + 1 = 0 \Rightarrow d = -1\)
4. From \(c + d = 0\): \(c - 1 = 0 \Rightarrow c = 1\)
5. From \(b + c = 0\): \(b + 1 = 0 \Rightarrow b = -1\)
6. From \(a + b = 0\): \(a - 1 = 0 \Rightarrow a = 1\)

### Step 6: Write the final partial fraction decomposition
Now we can substitute the values of \(a, b, c, d, e, f, g\) back into the partial fraction decomposition:

\[ \frac{1}{x^6(x+1)} = \frac{1}{x+1} - \frac{1}{x} + \frac{1}{x^2} - \frac{1}{x^3} + \frac{1}{x^4} - \frac{1}{x^5} + \frac{1}{x^6} \]

### Final Answer:
Thus, the partial fraction decomposition of \(\frac{1}{x^6(x+1)}\) is:

\[ \frac{1}{x+1} - \frac{1}{x} + \frac{1}{x^2} - \frac{1}{x^3} + \frac{1}{x^4} - \frac{1}{x^5} + \frac{1}{x^6} \]

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