SEQUENCE CONVERGENCE When all other methods fail, can you solve a limit of a sequence problem by substituting large values of n into the formula for the nth term?
a. It’s never a bad idea to look at the terms of a sequence for large values of n; the specific values you see may give you some insight into the nature of the sequence and help you formulate reasoning about the infinitely many terms you can’t see. The specific numbers may also give you more confidence – or sometimes less confidence – in the more abstract reasoning you do about the sequence. The specific numbers may convince you of something about the sequence, although even if you’re conced of the truth of something, the numbers themselves don’t provide understanding of why something is true. b. There are sequences which we can prove converge to a limit L, but we have no methods to determine the exact value of L. An approximate value for the limit is then a solution of sorts, especially if we can say something like, ” The limit is approximately 4.371, with error no larger than 0.0005.
a. It’s never a bad idea to look at the terms of a sequence for large values of n; the specific values you see may give you some insight into the nature of the sequence and help you formulate reasoning about the infinitely many terms you can’t see. The specific numbers may also give you more confidence – or sometimes less confidence – in the more abstract reasoning you do about the sequence. The specific numbers may convince you of something about the sequence, although even if you’re conced of the truth of something, the numbers themselves don’t provide understanding of why something is true. b. There are sequences which we can prove converge to a limit L, but we have no methods to determine the exact value of L.
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