Final answer:
To express a series as an alternating series, include a factor of (-1)⁻k in the general term, where the positive part of the general term is the term without the alternating factor.
Step-by-step explanation:
To write a given series as an alternating series in the form ∑ (-1)⁻k bₖ from k=0 to infinity, we assume that the series has terms which alternate in sign. The positive part of the general term, denoted by bₖ, would be the absolute value of the series' terms without consideration of their alternating signs (i.e., the sequence of positive numbers that you get if you were to remove the sign-changing factor).
If we are given a series such as a_n = n^2, for example, to make it into an alternating series, we would then write it as a_n = (-1)^n * n^2, where the positive part of the general term is simply the n^2, and the (-1)^n dictating the alternating pattern.
The student is asking to write a given series in part (a) as an alternating series in the form ∑(â???1)kbk and identify the positive part of the general term.
In order to write the series as an alternating series, we need to determine the alternating sign, which is (-1)^k. The positive part of the general term is represented by b_k. Therefore, the series can be written as ∑(â???1)kbk, where (-1)^k represents the alternating sign and b_k represents the positive part of the general term.