Examples of monotone in the following topics:
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- The theorem known as the "Leibniz Test," or the alternating series test, tells us that an alternating series will converge if the terms an converge to 0 monotonically.
- Proof: Suppose the sequence an converges to 0 and is monotone decreasing.
- Since an is monotonically decreasing, the terms are negative.
- an=n1 converges to 0 monotonically.
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- Consider an integer N and a non-negative function f defined on the unbounded interval [N,∞), on which it is monotonically decreasing.
- The above examples involving the harmonic series raise the question of whether there are monotone sequences such that f(n) decreases to 0 faster than n1but slower than n1+ε1 in the sense that:
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- Integral test: For a positive, monotone decreasing function f(x) such that f(n)=an, if ∫1∞f(x)dx=limt→∞∫1tf(x)dx<∞ then the series converges.