¿Qué número sigue? 1; 1; 3; 15; 105;
1 answer:
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Answer:
See answer below
Step-by-step explanation:
Define the intervals:
for n≥1.
The intervals are nested, in the sense that To see this, use the fact that for all n≥1, -1/n≤-1/(n+1) and 1/n≥1/(n+1) (intuitively, the intervals are "shrinking" in size, and are centered around √2).
The only point common to all intervals is √2, in the sense that
The idea for this construction is to center the intervals around √2 and shrink their size with the summand 1/n. As n goes to infinity, 1/n tends to zero and the intervals became closer and closer to √2, but they NEVER degenerate to the point √2, in contrast to their intersection.
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