To approximate definite integrals using series, you can use a technique called the "trapezoidal rule." This method involves dividing the interval of integration into smaller subintervals and using the sum of the areas of trapezoids to approximate the area under the curve. The more subintervals you use, the more accurate your approximation will be.
To apply this method to your specific example, you can divide the interval from 0 to 1 into smaller subintervals (such as 0 to 0.25, 0.25 to 0.5, 0.5 to 0.75, and 0.75 to 1) and use the trapezoidal rule to approximate the integral on each subinterval. Then, you can add up these approximations to get an overall approximation for the integral from 0 to 1.
To determine the accuracy of your approximation, you can use the remainder estimate for the integral test. This estimate tells you how close your approximation is to the actual value of the integral. If the remainder estimate is less than 0.001, then your approximation is within the desired accuracy.
Keep in mind that the remainder estimate for the integral test applies to both indefinite and definite integrals. In this case, you are using it to determine the accuracy of your approximation for the definite integral.
In summary, to approximate definite integrals using series, you can use the trapezoidal rule and the remainder estimate for the integral test. By dividing the interval of integration into smaller subintervals, you can improve the accuracy of your approximation. And by using the remainder estimate, you can determine if your approximation is within the desired accuracy.
