1 Consider the series ) n=1 a) Use the Remainder Estimate for the Integral Test to find an upper bound for the error in using S10 (the 10th partial sum) to approximate the sum of this series. b) How many terms suffice to ensure that the sum is accurate to within 10-6?
The Correct Answer and Explanation is :
To address your questions regarding the error estimation of a series using the Remainder Estimate for the Integral Test, let’s break down the process into two parts:
a) Upper Bound for the Error Using S₁₀
The Remainder Estimate for the Integral Test provides an upper bound for the error when approximating the sum of a convergent series by its nth partial sum. Specifically, if SS denotes the sum of the infinite series and SnS_n represents the nth partial sum, the error RnR_n is given by:
∣S−Sn∣≤∫n+1∞f(x) dx|S – S_n| \leq \int_{n+1}^{\infty} f(x) \, dx
where f(x)f(x) is a continuous, positive, and decreasing function that matches the terms of the series, i.e., f(n)=anf(n) = a_n.
To apply this to your series, we need to:
- Identify the function f(x)f(x): Determine the function whose terms correspond to the series terms. For example, if the series is ∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2}, then f(x)=1x2f(x) = \frac{1}{x^2}.
- Compute the integral: Evaluate the integral from n+1n+1 to infinity: ∫n+1∞f(x) dx\int_{n+1}^{\infty} f(x) \, dx This integral provides the upper bound for the error.
For instance, if f(x)=1x2f(x) = \frac{1}{x^2}, the integral becomes:
∫n+1∞1×2 dx=[−1x]n+1∞=1n+1\int_{n+1}^{\infty} \frac{1}{x^2} \, dx = \left[ -\frac{1}{x} \right]_{n+1}^{\infty} = \frac{1}{n+1}
Thus, the error when approximating the sum by the first 10 terms (i.e., S10S_{10}) is at most 111\frac{1}{11}.
b) Number of Terms for Desired Accuracy
To ensure that the error is less than a specified value, say ϵ=10−6\epsilon = 10^{-6}, we set up the inequality:
∫n+1∞f(x) dx≤ϵ\int_{n+1}^{\infty} f(x) \, dx \leq \epsilon
Solving this inequality will give the smallest integer NN such that the error is within the desired tolerance.
Continuing with the example where f(x)=1x2f(x) = \frac{1}{x^2}, we solve:
1n+1≤10−6\frac{1}{n+1} \leq 10^{-6}
This simplifies to:
n+1≥106n+1 \geq 10^6
Therefore, n≥999,999n \geq 999,999.
Thus, summing the first 999,999 terms ensures that the error is less than 10−610^{-6}.
Explanation
The Remainder Estimate for the Integral Test is a powerful tool for approximating the sum of a convergent series. By comparing the tail of the series to an improper integral, we can establish an upper bound for the error introduced by truncating the series after a finite number of terms. This method is particularly useful when the exact sum of the series is difficult to compute directly.
The key steps involve:
- Identifying the function f(x)f(x): This function should match the terms of the series and be continuous, positive, and decreasing for all xx greater than some integer NN.
- Evaluating the integral: Computing the integral from n+1n+1 to infinity provides the error bound. The choice of nn depends on the desired accuracy.
By applying these steps, we can determine how many terms are necessary to achieve a specific level of precision in approximating the sum of the series. This approach is widely used in mathematical analysis and numerical methods to assess the accuracy of series approximations.