Math 2.0

Sequences, series, summability Example

The sequence r^n tends to infinity for r > 1

Builds on Geometric sequenceSequence tending to infinity

Example

Let rr be a real number with r>1r > 1. Then rn→+∞r^n \to +\infty as n→∞n \to \infty, in the sense of a sequence tending to infinity. Write r=1+hr = 1 + h with h>0h > 0. By Bernoulli's inequality, rn≥1+nhr^n \ge 1 + n h for each natural number nn. Let MM be a real number, and choose a natural number NN with Nh>MN h > M. Then rn>Mr^n > M for all n≥Nn \ge N.

Remarks

For r<−1r < -1 the sequence rnr^n does not tend to +∞+\infty or to −∞-\infty. Its terms change sign, but ∣rn∣→+∞|r^n| \to +\infty. Exponential growth is faster than polynomial growth. The node exponentials dominate powers in sequences makes this precise.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionGeometric sequence, definition, no evidence levelGeometric sequencedefinitionSequence tending to infinity, definition, no evidence levelSequence tending to i…definitionThe sequence r^n tends to infinity for r > 1, example, no evidence levelThe sequence r^n tend…example
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  1. Sequence definition
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  4. Sequence tending to infinity definition
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Further reading

  1. Web https://en.wikipedia.org/wiki/Geometric_progression, Properties
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