Using Mathematical Induction to find the Limit of a Recursive Sequence

In this video I go over an example of a sequence that is defined by a recurrence relationship, and then use mathematical induction to prove it is increasing, bounded, thus convergent, and finally its actual limit. The first term of the sequence is 2, and the next terms are governed by the recurrence relation 1/2 (previous term + 6). Calculating these out manually suggests the terms are approaching the number 6. This suggests we use Mathematical induction by assuming the that terms are increasing and thus proving they are when starting with the initial term. I prove the sequence is bounded and increasing, hence the monotonic sequence theorem states it is convergent, but this doesn't tell us the value of the limit. I determine the limit using the recurrence relation in the limit as the terms go to infinity, and finally obtain the answer, which is the limit is 6. Amazing stuff!

#math #sequences #mathematicalinduction #calculus #education

Timestamps:

- Example 13: Recursive sequence: 0:00
- Initial terms suggest sequence is approaching 6: 2:44
- Recall mathematical induction: 3:03
- Applying mathematical induction to show sequence is increasing: 4:16
- Sequence is increasing by induction: 7:00
- Verifying the sequence is bounded, also using induction: 7:18
- Monotonic sequence states bounded, monotonic sequence is convergent: 8:58
- Determining the limit using the recurrence relation: 9:20
- Limit is six: 12:00

Notes and playlists:

- Summary: https://inleo.io/threads/view/mes/re-leothreads-m2rewifu
- Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HUgfK34Voi6BvrNqt4X4BV
- Notes: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more
- Infinite Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz .

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