## The challenge

Consider the sequence `U(n, x) = x + 2x**2 + 3x**3 + .. + nx**n` where x is a real number and n a positive integer.

When `n` goes to infinity and `x` has a correct value (ie `x` is in its domain of convergence `D`), `U(n, x)` goes to a finite limit `m` depending on `x`.

Usually given `x` we try to find `m`. Here we will try to find `x` (x real, 0 < x < 1) when `m` is given (m real, m > 0).

Let us call `solve` the function `solve(m)` which returns `x` such as U(n, x) goes to `m` when `n` goes to infinity.

#### Examples:

`solve(2.0) returns 0.5` since `U(n, 0.5)` goes to `2` when `n` goes to infinity.

`solve(8.0) returns 0.7034648345913732` since `U(n, 0.7034648345913732)` goes to `8` when `n` goes to infinity.

#### Note:

You pass the tests if `abs(actual - expected) <= 1e-12`

## The solution in Kotlin code

Option 1:

 ``````1 2 3 4 5 6 `````` ``````package solv fun solve(m:Double):Double { val s = Math.sqrt(4 * m + 1) return (2 * m + 1 - s) / (2 * m) } ``````

Option 2:

 ``````1 2 3 4 5 `````` ``````package solv import kotlin.math.sqrt fun solve(m: Double): Double = 1 + (0.5 - sqrt(0.25 + m)) / m ``````

Option 3:

 ``````1 2 3 4 5 `````` ``````package solv fun solve(s:Double):Double { return (1 - Math.sqrt(4 * s + 1)) / (2.0 * s) + 1; } ``````

## Test cases to validate our solution

 `````` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 `````` ``````package solv import org.junit.Assert.* import org.junit.Test import java.util.Random class solvTest { private fun assertFuzzy(m:Double, expect:Double) { val merr = 1e-12 println("Testing " + m) val actual = solve(m) println("Actual: " + actual) println("Expect: " + expect) val inrange = Math.abs(actual - expect) <= merr if (inrange == false) { println("Expected must be near " + expect + ", got " + actual) } println("-") assertEquals(true, inrange) } @Test fun test1() { assertFuzzy(2.00, 5.000000000000e-01) assertFuzzy(4.00, 6.096117967978e-01) assertFuzzy(5.00, 6.417424305044e-01) } } ``````