Error bounds for series practice problems
How many terms are enough? Error bounds from the alternating series test and the integral test. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
How many terms of \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{n^{2}} \) guarantee an error below \( \displaystyle \frac{1}{1000} \)? Use the alternating series estimate \( \displaystyle |R_N| \le b_{N+1} \).
Use the integral test bound \( \displaystyle R_N \le \int_N^\infty f(x)\,dx \) to find the smallest \( \displaystyle N \) for which the partial sum \( \displaystyle S_N \) of \( \displaystyle \sum 1/n^{2} \) is within \( \displaystyle \frac{1}{1000} \) of the sum.
How many terms of \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{n!} \) guarantee an error below \( \displaystyle \frac{1}{100} \)? Use the alternating series estimate \( \displaystyle |R_N| \le b_{N+1} \).
How many terms of \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{n^{3}} \) guarantee an error below \( \displaystyle \frac{1}{1000} \)? Use the alternating series estimate \( \displaystyle |R_N| \le b_{N+1} \).
How many terms of \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{n!} \) guarantee an error below \( \displaystyle \frac{1}{10000} \)? Use the alternating series estimate \( \displaystyle |R_N| \le b_{N+1} \).
Use the integral test bound \( \displaystyle R_N \le \int_N^\infty f(x)\,dx \) to find the smallest \( \displaystyle N \) for which the partial sum \( \displaystyle S_N \) of \( \displaystyle \sum 1/n^{4} \) is within \( \displaystyle \frac{1}{1000} \) of the sum.
How many terms of \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{n^{2}} \) guarantee an error below \( \displaystyle \frac{1}{10000} \)? Use the alternating series estimate \( \displaystyle |R_N| \le b_{N+1} \).
Use the integral test bound \( \displaystyle R_N \le \int_N^\infty f(x)\,dx \) to find the smallest \( \displaystyle N \) for which the partial sum \( \displaystyle S_N \) of \( \displaystyle \sum 1/n^{4} \) is within \( \displaystyle \frac{1}{100} \) of the sum.
Use the integral test bound \( \displaystyle R_N \le \int_N^\infty f(x)\,dx \) to find the smallest \( \displaystyle N \) for which the partial sum \( \displaystyle S_N \) of \( \displaystyle \sum 1/n^{3} \) is within \( \displaystyle \frac{1}{1000} \) of the sum.
Use the integral test bound \( \displaystyle R_N \le \int_N^\infty f(x)\,dx \) to find the smallest \( \displaystyle N \) for which the partial sum \( \displaystyle S_N \) of \( \displaystyle \sum 1/n^{3} \) is within \( \displaystyle \frac{1}{100} \) of the sum.