If `sqrt(1+1/(1^2)+1/(2^2))+sqrt(1+1/(2^2)+1/(3^2))+sqrt(1+1/(3^2)+1/(4^2))++sqrt(1+1/((1999)^2)+1/(2000)^2)+=1/x ,`Find x

Answer

To solve the given problem, we need to evaluate the expression:

\[ \sqrt{1 + \frac{1}{1^2} + \frac{1}{2^2}} + \sqrt{1 + \frac{1}{2^2} + \frac{1}{3^2}} + \sqrt{1 + \frac{1}{3^2} + \frac{1}{4^2}} + \ldots + \sqrt{1 + \frac{1}{1999^2} + \frac{1}{2000^2}} = \frac{1}{x} \]

### Step 1: Simplify Each Term
Let's denote the general term as:

\[ T_r = \sqrt{1 + \frac{1}{r^2} + \frac{1}{(r+1)^2}} \]

We will simplify \(T_r\):

\[ T_r = \sqrt{1 + \frac{1}{r^2} + \frac{1}{(r+1)^2}} = \sqrt{1 + \frac{(r+1)^2 + r^2}{r^2(r+1)^2}} \]

### Step 2: Find a Common Denominator
The common denominator for the fractions inside the square root is \(r^2(r+1)^2\):

\[ T_r = \sqrt{1 + \frac{(r^2 + 2r + 1 + r^2)}{r^2(r+1)^2}} = \sqrt{1 + \frac{2r^2 + 2r + 1}{r^2(r+1)^2}} \]

### Step 3: Combine Terms
Now, we can express \(T_r\) as:

\[ T_r = \sqrt{\frac{r^2(r+1)^2 + 2r^2 + 2r + 1}{r^2(r+1)^2}} = \sqrt{\frac{(r^2 + r + 1)^2}{r^2(r+1)^2}} \]

### Step 4: Simplify Further
Taking the square root:

\[ T_r = \frac{r^2 + r + 1}{r(r+1)} \]

### Step 5: Rewrite the Sum
Now we need to sum \(T_r\) from \(r = 1\) to \(1999\):

\[ \sum_{r=1}^{1999} T_r = \sum_{r=1}^{1999} \frac{r^2 + r + 1}{r(r+1)} \]

### Step 6: Break Down the Fraction
We can break down the fraction:

\[ \frac{r^2 + r + 1}{r(r+1)} = \frac{1}{r} + \frac{1}{r+1} + \frac{1}{r(r+1)} \]

### Step 7: Evaluate the Sum
Now, we can evaluate the sum:

\[ \sum_{r=1}^{1999} \left( \frac{1}{r} + \frac{1}{r+1} + \frac{1}{r(r+1)} \right) \]

The first two sums will telescope, and the last term can be evaluated separately.

### Step 8: Calculate the Total
The telescoping nature will lead to:

\[ \sum_{r=1}^{1999} \left( \frac{1}{r} + \frac{1}{r+1} \right) = 2 + \frac{1}{2000} - 1 = 1999 + \frac{1}{2000} \]

### Step 9: Set Equal to \(\frac{1}{x}\)
Now we have:

\[ \sum_{r=1}^{1999} T_r = 2000 - \frac{1}{2000} = \frac{2000^2 - 1}{2000} \]

Setting this equal to \(\frac{1}{x}\):

\[ \frac{2000^2 - 1}{2000} = \frac{1}{x} \]

### Step 10: Solve for \(x\)
Cross-multiplying gives:

\[ 2000^2 - 1 = \frac{2000}{x} \]

Thus,

\[ x = \frac{2000}{2000^2 - 1} \]

### Final Answer
The value of \(x\) is:

\[ x = \frac{2000}{2000^2 - 1} \]