If the sum ` sum_(n=1)^10 sum_(m=1)^10 tan^(-1) (m/n) = k pi`, find the value of k

Answer

To solve the problem, we need to evaluate the double summation:

\[ \sum_{n=1}^{10} \sum_{m=1}^{10} \tan^{-1}\left(\frac{m}{n}\right) \]

We can use the property of the inverse tangent function, which states:

\[ \tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}\left(\frac{x+y}{1-xy}\right) \quad \text{if } xy < 1 \]

and if \( xy = 1 \):

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

### Step 1: Rearranging the Summation

We can rewrite the double summation:

\[ \sum_{n=1}^{10} \sum_{m=1}^{10} \tan^{-1}\left(\frac{m}{n}\right) = \sum_{n=1}^{10} \left( \tan^{-1}\left(\frac{1}{n}\right) + \tan^{-1}\left(\frac{2}{n}\right) + \ldots + \tan^{-1}\left(\frac{10}{n}\right) \right) \]

### Step 2: Evaluating Inner Summation

For a fixed \( n \), we evaluate the inner summation:

\[ S_n = \sum_{m=1}^{10} \tan^{-1}\left(\frac{m}{n}\right) \]

### Step 3: Pairing Terms

Notice that:

\[ \tan^{-1}\left(\frac{m}{n}\right) + \tan^{-1}\left(\frac{n}{m}\right) = \frac{\pi}{2} \quad \text{for } m \neq n \]

This means we can pair terms in the summation. For each \( n \), we can pair \( m \) and \( 11-m \) (for \( m = 1, 2, \ldots, 10 \)).

### Step 4: Counting the Pairs

For \( n = 1 \) to \( 10 \), there are \( 10 \) terms in the inner summation, and we can form \( 5 \) pairs of \( \tan^{-1}\left(\frac{m}{n}\right) + \tan^{-1}\left(\frac{n}{m}\right) \) for \( m = 1, 2, \ldots, 10 \).

### Step 5: Total Contribution from Pairs

Each pair contributes \( \frac{\pi}{2} \), and since there are \( 5 \) pairs for each \( n \):

\[ S_n = 5 \cdot \frac{\pi}{2} = \frac{5\pi}{2} \]

### Step 6: Summing Over \( n \)

Now, we sum \( S_n \) over \( n \):

\[ \sum_{n=1}^{10} S_n = \sum_{n=1}^{10} \frac{5\pi}{2} = 10 \cdot \frac{5\pi}{2} = 25\pi \]

### Step 7: Relating to \( k \)

The original problem states:

\[ \sum_{n=1}^{10} \sum_{m=1}^{10} \tan^{-1}\left(\frac{m}{n}\right) = k\pi \]

From our calculation, we found that the total sum equals \( 25\pi \). Therefore, we can conclude that:

\[ k = 25 \]

### Final Answer

Thus, the value of \( k \) is:

\[ \boxed{25} \]