Estimate the error in approximating the sum of the alternating series $S=\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{1+2^{n}}$ by the sum of the first 11 terms.
A)0.000024
B)0.00024
C)0.0024
D)0.024

Question

Quizplus · Accepted Answer

The error in approximating the sum of an alternating series by the sum of its first $N$ terms is less than or equal to the absolute value of the first omitted term. In this case, the first omitted term after summing the first 11 terms is the 12th term, which is $\frac{(-1)^{12-1}}{1+2^{12}} = \frac{1}{1+4096}$. This value is approximately $0.00024$, making option B the correct choice.

Estimate the Error in Approximating the Sum of the Alternating S=∑n=1∞(−1)n−11+2nS=\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{1+2^{n}}S=∑n=1∞​1+2n(−1)n−1​

Estimate the Error in Approximating the Sum of the Alternating $S=\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{1+2^{n}}$