Subsequential Limit

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A subsequential limit is the limit of a subsequence; a subsequence is a smaller part of some larger sequence. For example, the integer sequence {1, 2, 1, 2} has two subsequences: one for odd numbers and one for even numbers.

Sometimes a limit doesn’t exist for a sequence, but does exist for one or more subsequences. For example, the limit for the following sequence doesn’t exist:

However, there are three subsequences with limits [1]:

• (x1, x4, x7, …) converges to 1.
• (x2, 51, 81, …) converges to 0.
• (x3, x6, x9, …) converges to -1.

Examples: Finding a Subsequential Limit

Example question #1: Find the subsequential limits of the sequence an = 1 – (-1)n.

The key here is to look for a pattern by generating some terms.

Step 1: Generate some terms of the sequence. One of the easiest ways to do this is in Excel. Here’s the steps:

1. Type the numbers 1 through 10 in column A of a worksheet.
2. Type the formula in cell B2. When you get to the “n”, click on cell A1.
3. Press Enter, then grab the little square at the right hand corner of B2. Drag this square to the bottom of column B2.
4. Step 2: Identify the pattern. We have two subsequences here: odd numbers where the limit is 2 and even numbers where the limit is 0. Therefore, the subsequential limits are 0 and 2.

Example question #2: Find the subsequential limits of the sequence an = 1 – (-1)n.

Step 1: Generate some terms of the sequence. For this example, I didn’t see a pattern for the first ten terms, so I went much larger:

Step 2: Identify the pattern.The sequence (and all of its subsequences) converge to 1. This is an example where it may be easier to demonstrate the limit with algebra. We know that |an| = 1/n → -, so an → 1.

The limits of subsequences can more formally be defined by the limit inferior and limit superior.

References

Upper and Lower Limits of Sequences of Real Numbers. https://faculty.math.illinois.edu/~r-ash/RV/RV3.pdf

CITE THIS AS:
Stephanie Glen. "Subsequential Limit" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/subsequential-limit/
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