Infinite Nested Roots

Infinite Nested Roots — Interactive Calculator

Choose the form of the nested expression and enter parameters. The page shows the algebraic formula (if available) and computes a numeric answer with steps.

Expression type √(x + √(x + √(x + ...))) — (square root, positive) √(x - √(x - √(x - ...))) — (square root, negative) n√( x * n√( x * n√( x * ... ))) — multiplicative pattern n√( x + n√( x + n√( x + ... ))) — additive pattern (general)

Value x

Root order n

Decimal places

Presets (quick):

Notes:

• For √(x + √(x + ...)) we use y = √(x + y) → y² - y - x = 0 → y = (1 + √(1+4x))/2.
• For √(x - √(x - ...)) we use y = √(x - y) → y² + y - x = 0 → y = (-1 + √(1+4x))/2.
• For multiplicative nested form n√(x * n√(x * ...)) we use y^n = x·y → y^{n-1} = x → y = x^{1/(n-1)}.
• For additive n-th root (y = n√(x + y)) we solve y^n - y - x = 0 numerically (positive real root).