Calculus II: Taylor & Maclaurin Series

                <h2>Approximating Functions</h2>
                <p>A Taylor series allows us to represent complex functions (like sin(x) or e^x) as an infinite sum of polynomial terms. This is how calculators compute Sine and Cosine!</p>

                <h3>General Formula (centered at a):</h3>
                <p>f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...</p>

                <h3>Maclaurin Series (centered at 0):</h3>
                <p>A special case where a = 0. Example for e^x:</p>
                <pre>e^x ≈ 1 + x + x^2/2! + x^3/3! + ...</pre>

                <h2>Ratio Test Application</h2>
                <p>Not all series converge for all x. To find the <b>Interval of Convergence</b>, we typically use the Ratio Test.</p>

                <h3>The Logic:</h3>
                <ol>
                    <li>Take the limit as n approaches infinity of |a_(n+1) / a_n|.</li>
                    <li>Set the result < 1 to find the values of x where it converges.</li>
                    <li><b>Check Endpoints:</b> Always plug the endpoints back into the original series to check for conditional convergence.</li>
                </ol>
                <p><b>Homework Help:</b> Use our <b>AI Math Solver</b> to check your ratio test steps if you keep getting 'Infinity'.</p>