Question

Explain why the sequence of partial sums for an alternating series is not an increasing sequence.

Short answer

Answer: The sequence of partial sums for an alternating series is not an increasing sequence because the terms in the series alternate between positive and negative values. As a result, the sequence of partial sums will also alternate between increasing and decreasing as terms are added, depending on the sign of the term being added.

Step-by-step solution

Define an alternating series and its sequence of partial sums

An alternating series is a series of the form: \[ \sum_{n=1}^{\infty} (-1)^{n-1} a_n \quad \text{or} \quad \sum_{n=1}^{\infty} (-1)^n a_n, \] where \( a_n \) are positive, real numbers. The sequence of partial sums is given by: \[ s_i = \sum_{n=1}^{i} (-1)^{n-1} a_n \quad \text{or} \quad s_i = \sum_{n=1}^{i} (-1)^n a_n, \] for all positive integers \( i \).

Examine the terms in the sequence of partial sums

Since the alternating series has both positive and negative terms, adding a term in the series can either increase or decrease the partial sums. We'll expand the sequence of partial sums for a few terms: In the first case, the series form: \[\sum_{n=1}^{\infty} (-1)^{n-1} a_n:\] \[s_1 = a_1, \] \[s_2 = a_1 - a_2, \] \[s_3 = a_1 - a_2 + a_3, \] \[\dots\] In the second case, the series form: \[\sum_{n=1}^{\infty} (-1)^n a_n:\] \[s_1 = -a_1, \] \[s_2 = -a_1 + a_2, \] \[s_3 = -a_1 + a_2 - a_3, \] \[\dots\] We can observe that in both cases, the sign of the terms changes in each successive partial sum.

Determine if the sequence of partial sums is increasing

Taking the series form \(\sum_{n=1}^{\infty} (-1)^{n-1} a_n\), we notice that the partial sums behave as follows: 1. When adding a positive term (even index): \(s_{i+1} = s_i + a_{i+1}\), the sum increases. 2. When adding a negative term (odd index): \(s_{i+1} = s_i - a_{i+1}\), the sum decreases. Now, consider the series form \(\sum_{n=1}^{\infty} (-1)^n a_n\): 1. When adding a negative term (odd index): \(s_{i+1} = s_i - a_{i+1}\), the sum decreases. 2. When adding a positive term (even index): \(s_{i+1} = s_i + a_{i+1}\), the sum increases. In either case, the sequence of partial sums is not consistently increasing or decreasing. Instead, the sequence alternates between increasing and decreasing as terms are added, depending on the sign of the term being added.

Conclusion

The sequence of partial sums for an alternating series is not an increasing sequence because the terms in the series alternate between positive and negative values. As a result, the sequence of partial sums will also alternate between increasing and decreasing as terms are added, depending on the sign of the term being added.