Problem 46 - Calculus: One and Several Variables

Question

Let \(\sum_{i=1}^{\infty}(-1)^{k} a_{k}\) be an alternating series with the \(a_{k}\) forming a decreasing sequence of positive numbers. Show that the sequence of partial sums of odd index increases and is bounded above.

Step-by-Step Solution

Answer

The sequence of partial sums of odd index of the given alternating series is non-decreasing and bounded above by \(a_{1}\).

Step 1: Understand Alternating Series

An alternating series is a series where the sign of the terms alternates between positive and negative. It can be represented as \(\sum_{i=1}^{\infty}(-1)^{k} a_{k}\), where \(a_{k}\) is a sequence of positive numbers.

Step 2: Defining partial sums and showing they are non-decreasing

For the given alternating series, partial sums of odd index can be represented as \(S_{2k-1} = a_{1} - a_{2} + a_{3} - a_{4} +....+ a_{2k-1}\). For \(S_{2k+1} = S_{2k-1} + a_{2k} - a_{2k+1}\), we can see that as \(a_{k}\) forms a decreasing sequence, \(a_{2k} - a_{2k+1} \geq 0\). Thus, \(S_{2k+1} \geq S_{2k-1}\) meaning the sequence of partial sums of odd index is non-decreasing.

Step 3: Showing that the sequence is bounded above

To show the sequence is bounded above, note that all terms in the sequence of partial sums are less than or equal to the first term, \(a_{1}\). Hence, the sequence is bounded above by \(a_{1}\).

Key Concepts

Partial Sums

In the context of series, a partial sum is the sum of the first few terms up to a specific point. For example, given the alternating series \( \sum_{i=1}^{\infty}(-1)^{k} a_{k} \), we can consider the partial sums of odd indices. These partial sums are represented as:

• \( S_{1} = a_{1} \)
• \( S_{3} = a_{1} - a_{2} + a_{3} \)
• \( S_{5} = a_{1} - a_{2} + a_{3} - a_{4} + a_{5} \)

Each partial sum adds more terms of the series, taking account of their alternating signs.
For partial sums of an odd index, we start at an odd position and include every second number. This means the sequence often includes more positive than negative numbers. Thus, it can be seen that these partial sums eventually form an increasing sequence.

Decreasing Sequence

A sequence is termed as decreasing if each subsequent term is less than or equal to its predecessor. In the context of the alternating series, we consider the sequence \( a_k \) to be decreasing. This is important because it inherently affects the behavior of the series' partial sums.
The decreasing nature of \( a_k \) suggests:

• \( a_{1} \geq a_{2} \geq a_{3} \geq \cdots \)

Knowing the sequence decreases helps us understand why terms like \( a_{2k} - a_{2k+1} \) are non-negative (or zero).
This ensures that when considering any two consecutive partial sums of odd indices, the latter is equal to or greater than the former, accounting for the non-decreasing nature of these sums.

Bounded Sequence

A sequence is said to be bounded above or below if there's a limit that it doesn't surpass. In the case of the partial sums of the given alternating series, we focus on them being bounded above.

All terms in the sequence of partial sums must be less than or equal to the first term, \( a_{1} \), in our example. This means no partial sum will exceed the size of the very first term in the sequence and hence ensures boundedness above.
The bounded nature can be visualized as follows:

• No matter how many terms we include in the sum, the total won't surpass \( a_{1} \).


• The presence of negative terms counterbalances the increase due to positive terms, contributing to this upper boundary.

This concept is a key player in analyzing convergence behavior in alternating series. A bounded sequence guarantees that even as more terms are added, infinite divergence is prevented.