![]() ![]() If r 1, the geometric sequence will be a sequence of identical constants, and is therefore trivial. There are many ways to determine if a sequence convergestwo are listed below. If a sequence is bounded but not monotonic, it might converge or it might diverge. A geometric sequence converges if -1 < r 1 and diverges if r -1 or r > 1. 4: Convergence of Sequences and Series 4. Sometimes the Squeeze Theorem can be rather useful provided we can find two other sequences that converge to the same limit for which our unknown sequence is.Unboundedness Theorem: If a sequence is not bounded, it diverges. A sequence is said to be convergent if it approaches some limit (DAngelo and West 2000, p. The key part of the following proof is the argument to show that a pointwise convergent, uniformly Cauchy sequence converges uniformly. Also, we prove the bounded monotone convergence theorem (BMCT), which asserts that every bounded monotone sequence is convergent.Monotonic Convergence Theorem: If a sequence is monotonic and bounded, if converges.If a sequence is either non-increasing or non-decreasing, it is called monotonic.Ī word of caution: The terms increasing and decreasing are dangerously ambiguous, since some authors use them to mean "strictly increasing" and "strictly decreasing", while others use them to mean "non-decreasing" and "non-increasing". In addition to certain basic properties of convergent sequences, we also study divergent sequences and in particular, sequences that tend to positive or negative innity. In other words, there is a bound $M$ such that every term in the sequence has size less than $M$.Ī sequence \le a_n$. 2 Sequences: Convergence and Divergence In Section 2.1, we consider (innite) sequences, limits of sequences, and bounded and monotonic sequences of real numbers. ![]() Improper Rational Functions and Long DivisionĬonvergence of Series with Negative Termsĭerivatives and Integrals of Power SeriesĪdding, Multiplying, and Dividing Power SeriesĪ sequence is bounded if $|a_n|$ never grows beyond a fixed size $M$. Indefinite Integrals and Anti-derivatives The Indefinite Integral and the Net Change If a sequence is undefined at a certain integer value e.g ((-1)n)/4-n2 is clearly undefined at n2. If the sequence of partial sums is a convergent sequence ( i.e. Therefore it may be convenient to think of sequences as the more natural idea, and series as a special case of sequences.The Indefinite Integral as Antiderivative We said that in order to determine whether a sequence 1anl converges or diverges, we need to examine its behaviour as n gets bigger and bigger. However, in the more general setting of topology the notion of a sequence is more general than the notion of a series, because in topological spaces without a vector space structure the notion of a series makes no sense (how will you define a series if you can't add things?). We know that $\lim_^N a_n = a_1 a_N - a_1 = a_N. The trouble is that the convergence of the terms tells you nothing about the convergence of the series.
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