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Chapter 7: Q. 19 (page 603)

Give examples of sequences satisfying the given conditions or explain why such an example cannot exist.

Complete Example 3 by showing that the limit of the sequencek3ekis0.

Short Answer

Expert verified

To prove that the limit of ak=k3ekis 0, first apply the L'Hospital's Rule and simply then again apply the L'Hospital's Rule and simply the following.

Step by step solution

01

Step 1. Given information.

Consider the given question,

The limit of the sequencek3ekis0.

02

Step 2. Find the limit.

Consider the sequence ak=k3ek.

The given sequence is bounded and eventually monotonic. Then the sequence is convergent.

The limit limk鈫�鈭�ak=limk鈫�鈭�k3ek.

The expression k3ekis indeterminate from of type 鈭�鈭�.

Applying limits,

limk鈫�鈭�ak=limk鈫�鈭�3k2ek=limk鈫�鈭�6kek=limk鈫�鈭�6ek=0

Hence, it has been proven that the limit of the sequenceak=k3ekis0.

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