explain this---arctan(x) = x - x^3/3 + x^5/5 - ...
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The arctangent function, written as arctan of x, is the inverse of the tangent function. Its domain includes all real numbers, while its range is bounded between negative pi over 2 and pi over 2. The graph shows this S-shaped curve approaching horizontal asymptotes. Our main question today is: how can we express this function as an infinite series expansion?
To find the series expansion of arctangent, we use Taylor series. The Taylor series expands any function around a point using its derivatives. The general formula shows how we build the series using the function value and all its derivatives at x equals zero. This is called a Maclaurin series. For our target function arctangent of x, we need to systematically calculate its derivatives and evaluate them at zero.
Now let's systematically calculate the derivatives of arctangent at x equals zero. Starting with the function itself, arctangent of zero equals zero. The first derivative is one over one plus x squared, which gives us one at x equals zero. The second derivative involves negative two x, giving zero at x equals zero. The third derivative gives us negative two. Continuing this pattern, we discover that non-zero derivatives occur only at odd orders with alternating signs, following a specific factorial pattern.
Now we substitute our calculated derivatives into the Taylor series formula. The constant term is zero since arctangent of zero equals zero. The first derivative gives us the x term. The second derivative is zero, so no x squared term. The third derivative of negative two gives us negative x cubed over three. The fifth derivative of twenty-four gives us x to the fifth over five. Continuing this pattern, we get the complete series: arctangent of x equals x minus x cubed over three plus x to the fifth over five minus x to the seventh over seven, and so on.
The arctangent series converges for absolute value of x less than or equal to one. Let's see a numerical example: arctangent of 0.5 using the first three terms gives approximately 0.4646, very close to the exact value of 0.4636. A famous application is the Leibniz formula for pi: pi over four equals arctangent of one, which gives us the alternating series one minus one-third plus one-fifth minus one-seventh and so on. The graph shows how partial sums converge to the exact function.