`x=sqrt(1+2sqrt(1+3sqrt(1+4sqrt(1+...`

Answer

To solve the problem \( x = \sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + 4\sqrt{1 + \ldots}}}} \), we will analyze the structure of the infinite nested radical.

### Step-by-step Solution:

1. **Assume the Value of \( x \)**:
We start by assuming that \( x \) is equal to the entire expression:
\[ x = \sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + 4\sqrt{1 + \ldots}}}} \]

2. **Square Both Sides**:
To eliminate the outer square root, we square both sides:
\[ x^2 = 1 + 2\sqrt{1 + 3\sqrt{1 + 4\sqrt{1 + \ldots}}} \]

3. **Isolate the Nested Radical**:
Rearranging gives us:
\[ x^2 - 1 = 2\sqrt{1 + 3\sqrt{1 + 4\sqrt{1 + \ldots}}} \]

4. **Divide by 2**:
Next, we divide both sides by 2:
\[ \frac{x^2 - 1}{2} = \sqrt{1 + 3\sqrt{1 + 4\sqrt{1 + \ldots}}} \]

5. **Square Again**:
We square both sides again to eliminate the square root:
\[ \left(\frac{x^2 - 1}{2}\right)^2 = 1 + 3\sqrt{1 + 4\sqrt{1 + \ldots}} \]

6. **Expand the Left Side**:
Expanding the left side gives:
\[ \frac{(x^2 - 1)^2}{4} = 1 + 3\sqrt{1 + 4\sqrt{1 + \ldots}} \]

7. **Isolate the Next Nested Radical**:
Rearranging gives us:
\[ \frac{(x^2 - 1)^2}{4} - 1 = 3\sqrt{1 + 4\sqrt{1 + \ldots}} \]

8. **Divide by 3**:
Dividing both sides by 3 gives:
\[ \frac{(x^2 - 1)^2 - 4}{12} = \sqrt{1 + 4\sqrt{1 + \ldots}} \]

9. **Continue This Process**:
This process can be repeated, but instead, we can recognize that this series converges to a specific value. Through historical mathematical insights, particularly from Ramanujan, we find that this infinite series converges to 3.

10. **Final Conclusion**:
Therefore, the value of \( x \) is:
\[ x = 3 \]