The value of ` Sigma_( r =2)^( infty) tan^(-1) ( 1/( r^(2) - 5r + 7)) ` , is

Options

A. `pi/4`
B. `pi/2`
C. `(3pi)/4`
D. `(5pi)/4`

Answer

To solve the problem, we need to evaluate the infinite series:

\[ \sum_{r=2}^{\infty} \tan^{-1} \left( \frac{1}{r^2 - 5r + 7} \right) \]

### Step 1: Simplify the Argument of the Inverse Tangent
First, we simplify the expression inside the series. The quadratic expression can be rewritten:

\[ r^2 - 5r + 7 = (r - 3)(r - 2) + 1 \]

This allows us to express the series as:

\[ \sum_{r=2}^{\infty} \tan^{-1} \left( \frac{1}{(r-3)(r-2) + 1} \right) \]

### Step 2: Use the Inverse Tangent Identity
We can use the identity for the difference of inverse tangents:

\[ \tan^{-1}(a) - \tan^{-1}(b) = \tan^{-1}\left( \frac{a-b}{1+ab} \right) \]

In our case, we set \( a = r - 2 \) and \( b = r - 3 \):

\[ \tan^{-1}(r - 2) - \tan^{-1}(r - 3) = \tan^{-1}\left( \frac{(r-2) - (r-3)}{1 + (r-2)(r-3)} \right) = \tan^{-1}\left( \frac{1}{(r-3)(r-2) + 1} \right) \]

### Step 3: Rewrite the Series
Thus, we can rewrite our series as:

\[ \sum_{r=2}^{\infty} \left( \tan^{-1}(r - 2) - \tan^{-1}(r - 3) \right) \]

### Step 4: Recognize the Telescoping Nature
This series is telescoping. When we expand it, we see:

\[ \left( \tan^{-1}(0) - \tan^{-1}(-1) \right) + \left( \tan^{-1}(1) - \tan^{-1}(0) \right) + \left( \tan^{-1}(2) - \tan^{-1}(1) \right) + \ldots \]

Most terms cancel out, leaving us with:

\[ \lim_{n \to \infty} \left( \tan^{-1}(n - 2) - \tan^{-1}(0) \right) \]

### Step 5: Evaluate the Limits
As \( n \to \infty \):

\[ \tan^{-1}(n - 2) \to \frac{\pi}{2} \]
and
\[ \tan^{-1}(0) = 0 \]

Thus, we have:

\[ \frac{\pi}{2} - \tan^{-1}(-1) = \frac{\pi}{2} + \frac{\pi}{4} \]

### Step 6: Final Calculation
Combining these results gives:

\[ \frac{\pi}{2} + \frac{\pi}{4} = \frac{2\pi}{4} + \frac{\pi}{4} = \frac{3\pi}{4} \]

### Conclusion
Therefore, the value of the infinite series is:

\[ \sum_{r=2}^{\infty} \tan^{-1} \left( \frac{1}{r^2 - 5r + 7} \right) = \frac{3\pi}{4} \]

### Final Answer
The correct option is \( \frac{3\pi}{4} \).

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