Taylor Coefficients of Natural Log Calculator

It starts with the zero constant coefficient, then applies the alternating reciprocal pattern: 1, -1/2, 1/3, -1/4, and so on. Every result identifies its power, numeric coefficient, and exact fraction, making the output useful for studying calculus, checking a hand-derived expansion, or feeding a finite logarithm approximation into software. The calculation is deterministic and supports degrees from zero through one thousand.

Choose the highest power included in the series

Enter a nonnegative integer called the degree. The calculator interprets that value inclusively, so degree 6 returns coefficients for x to the powers 0, 1, 2, 3, 4, 5, and 6. Degree 0 is valid and returns only the constant coefficient, which is zero because ln(1+0)=0. Degree 1 returns the linear truncation x. Keeping the degree separate from the number of nonzero terms prevents an easy off-by-one mistake: the output always has degree plus one coefficient records because the constant term is represented explicitly. The accepted range ends at 1000, which is ample for coefficient generation while keeping the response bounded and practical to inspect. Do not enter a decimal, numeric string, negative value, infinity, or omitted field; the input must be an actual integer in the declared range. This tool generates coefficients rather than evaluating the polynomial at a particular x, so there is no x input. Once the records are returned, you can use as many of them as your approximation, proof, worksheet, or program requires. Identical degrees always produce identical ordered output.

Read the alternating reciprocal coefficients

The Maclaurin expansion is ln(1+x)=x-x^2/2+x^3/3-x^4/4+..., so the coefficient of x^n is (-1)^(n+1)/n for every integer n at least 1. The constant coefficient at power 0 is handled separately as zero. Each returned record contains power, coefficient, and exact. Power is the exponent n. Coefficient is the JavaScript numeric value, convenient for direct computation. Exact is a string such as 1/3 or -1/4, preserving the rational form without floating-point display ambiguity; the first coefficient is written simply as 1. Odd positive powers have positive coefficients, while even positive powers have negative coefficients. The records are ordered from power 0 through the selected degree, so their array position also matches the exponent. The response additionally names the function, gives the expansion point as zero, and reports a radius of convergence of one. That radius describes the power series centered at zero, not a guarantee that a finite truncation is accurate to a particular tolerance. No symbolic algebra system or numerical differentiation is involved: the calculator applies the known coefficient formula directly, which makes even large requested degrees predictable and easy to audit.

Use a finite Taylor polynomial responsibly

A returned coefficient list defines a polynomial approximation when you multiply each coefficient by x raised to its recorded power and add the terms. For values with absolute value below one, increasing the degree generally improves the approximation, although convergence becomes slow as x approaches either boundary. At x=1, the infinite series becomes the alternating harmonic series and converges to ln(2), but a finite result still has truncation error. At x=-1 the logarithm is not finite, and the corresponding series does not provide a finite function value. Outside the radius of convergence, adding more Taylor terms does not turn this expansion into a valid logarithm approximation. The calculator intentionally does not evaluate x, estimate the remainder, select a degree for a requested tolerance, simplify a user-supplied function, or expand ln(a+x) around another center. Its narrow purpose is to return the coefficients of ln(1+x) accurately and consistently. That makes it well suited to classroom demonstrations, unit-test fixtures, and numerical code that performs its own evaluation and error control. Browser use provides an immediate result, while an API request costs $0.002 and returns the same structured coefficient data for automated workflows.