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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.

ekk

Short Answer

Expert verified

The sequence is an increasing function with no upper bounds and the limit of the sequence does not exist.

Step by step solution

01

Step 1. Given Information

The given sequence isekk.

02

Step 2. Use Derivative Test

  • Consider the function exx.
  • Its derivative is ex(x-1)x2.
  • The derivative is positive when x>1.
  • So, the function is an increasing function.
03

Step 3. Check Boundedness and Convergence

  • The derivative test tells that the function has the minimum value at x=1.
  • However, the function has no upper bounds.
  • An unbounded function is non-convergent and hence, the limit of the sequence does not exist.

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