Repeating Decimals and Infinite Geometric Series

Tap a term to see what it means, or .

Calculate

Example: what is 0.999…, with the 9 repeating forever? Try also 0.333… (block 3), 0.1666… (non-repeating 1, block 6) or 2.0909… (whole part 2, block 09 — in the block, the leading zero counts).

Whole part Before the decimal point. Non-repeating digits (optional) The ones that don't repeat. Repeating block The ones that repeat.

Step by step

What each term means

Sum of the Infinite Series (S): The value of a + ar + ar² + ⋯ with infinitely many terms. It exists when |r| < 1: the terms shrink so fast that the sum stops growing.
First Term (a): In a repeating decimal, the first copy of the block in its decimal place: in 0.333…, a = 3/10.
One Minus the Ratio (1 − r): Each repetition of the block is 10ᵏ times smaller, with k digits in the block: r = 1/10ᵏ. In 0.333…, r = 1/10, and S = (3/10)/(9/10) = 1/3.