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Limits of sequences: Determine whether the sequences that follow are bounded, monotonic and/or eventually monotonic.

Determine whether each sequence converges or diverges. If the sequence converges, find its limit.

k!1·3·5···(2k-1)

Short Answer

Expert verified
  • The sequence is montonically decresing and bounded.
  • The limit of the sequence is 0.

Step by step solution

01

Step 1. Given Information

The given sequence is k!1·3·5···(2k-1).

02

Step 2. Apply Ratio Test

  • Find the ratio ak+1ak.

(k+1)!1·3·5···(2k-1)(2k+1)k!1·3·5···(2k-1)=(k+1)!1·3·5···(2k-1)(2k+1)×1·3·5···(2k-1)k!=k+12k+1

  • As k+12k+1<1 for all k, the sequence is strictly monotonically decreasing.
03

Step 3. Check for Boundedness and Limit

  • All the terms of the sequence are positive and the sequence is decreasing, it is bounded below by zero, and because the sequence is strictly decreasing, the first term, 1 will be the upper bound for the sequence.
  • Therefore, the sequence is bounded.
  • The limit of the sequence will be 0, that is the lower bound of the decreasing sequence,

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