Sequence Math Example 4

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Example 4

medium
Determine whether an=2nn!a_n = \frac{2^n}{n!} converges or diverges.

Solution

  1. 1
    Compare consecutive terms: an+1an=2n+1\frac{a_{n+1}}{a_n} = \frac{2}{n+1}.
  2. 2
    As n→∞n \to \infty, 2n+1→0\frac{2}{n+1} \to 0.
  3. 3
    Since each term becomes a vanishing fraction of the previous, an→0a_n \to 0.

Answer

The sequence converges to 00.
Factorial grows far faster than any exponential. The ratio of successive terms going to 0 confirms the sequence collapses to 0.

About Sequence

An ordered list of numbers generated by a rule, where each number has a specific position (first, second, third, ...).

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More Sequence Examples

Example 1 easy

Write the first five terms of the sequence [formula] for [formula]

Example 2 medium

Determine whether the sequence [formula] converges or diverges. If it converges, find the limit.

Example 3 easy

Write the first four terms of [formula].

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