Muddiest Point 9/19/2017

Chapter 2 introduces sequences, series, and several theorems for proving the convergence or divergence for each. However, the theorems cannot be used interchangeably for sequences and series. This Muddiest Point post will distinguish between some theorems for sequences and series.

Convergence Theorems for Sequences

  • The Monotone Convergence Theorem: If a sequence is monotone and bounded, then it converges.
  • Theorem 2.5.2: Subsequences of a convergent sequence converge to the same limit as the original sequence.
  • Bolzano-Weierstrass Theorem: Every bounded sequence contains a convergent subsequence.
  • Cauchy Criterion: A sequence converges if and only if it is a Cauchy Sequence.
  • We showed in challenge 4 of homework 1 that if (b_n)\rightarrow b, then |b_n|\rightarrow |b|.

Convergence Theorems for Series

  • The Cauchy Condensation Test: Suppose b_n is decreasing and satisfies b_n \geq 0 for all n\in N. Then the series \Sigma _{n=1} ^\infty b_n converges if and only if the series \Sigma _{n=1} ^\infty 2^n b_{2^n} converges.
  • The Cauchy Criterion for Series: The series \Sigma _{k=1} ^\infty a_k converges if and only if, given \epsilon >0, there exists an N\in \mathbb{N} such that whenever n>m\geq N it follows that |a_{m+1}+ a_{m+2}+...+a_n|<\epsilon.
  • Theorem 2.7.3: If the series\Sigma _{k=1} ^\infty a_k \Sigma _{k=1} ^\infty a_k converges, then (a_k)\rightarrow 0.
  • Comparison Test: Assume (a_k) and (b_k) are sequences satisfying 0\leq a_k\leq b_k for all k\in \mathbb{N} i) If \Sigma _{k=1} ^\infty b_k converges, then \Sigma _{k=1} ^\infty a_k converges. ii) If \Sigma _{k=1} ^\infty a_k diverges, then \Sigma _{k=1} ^\infty B_k diverges.
  • Absolute Convergence Test: If the series \Sigma _{n=1} ^\infty |a_n| converges, then \Sigma _{n=1} ^\infty a_n converges as well.
  • Alternating Series Test: Let (a_n) be a sequence satisfying (i) a_1\geq a_2\geq a_3\geq...\geq a_n\geq a_{n+1}\geq ... and (ii)(a_n)\rightarrow 0. Then the alternating series \Sigma _{n=1} ^\infty (-1)^{n+1} a_n converges.

Note that while these theorems apply specifically to sequences or series, the convergence of the series \Sigma _{k=1} ^\infty a_k is defined in terms of the sequence of partial sums (s_n). Thus, you can also show the convergence of a series by identifying a formula and showing convergence for its sequence of partial sums.

Posted on by Elaine Wissuchek
Categories: Muddiest Point

One Response

  1. Jeremy LeCrone says:
    September 23, 2017 at 8:18 pm

    Thank you for this clarification of a very important distinction between sequences and series. It is easy to confuse these two concepts, but it is very critical that everyone pay careful attention to the results that apply in each setting and how to mentally distinguish them.

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