Calculus How To

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:
sequence with no limit

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. generating terms of a sequence in excel
  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:
    example 2 subsequence limit

    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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