Series with Arbitrary Signs

From the study guide

Course Outline

  1. Alternating Series and Their Terms
  2. Alternating Series Test
  3. Alternating Riemann Series
  4. Absolute Convergence
  5. Conditional Convergence
  6. Decision Strategy for Signed Series

1. Alternating Series and Their Terms

Key Concepts & Definitions

  • Alternating series: A real series whose terms can be written as un=(βˆ’1)nanu_n=(-1)^n a_n or un=(βˆ’1)n+1anu_n=(-1)^{n+1}a_n for a positive sequence ana_n uniquely given by an=∣un∣a_n=|u_n|.

Essential Points

  • In an alternating series, the signs of consecutive terms alternate, so even-indexed and odd-indexed terms have opposite signs.

2. Alternating Series Test

β˜… Must-know

πŸ“Œ The Alternating Series Test states that a series un=(βˆ’1)nanu_n=(-1)^n a_n converges if anβ‰₯0a_n\geq 0 is eventually decreasing and anβ†’0a_n\to0.

Further detail

  • For a convergent alternating series, the remainder after the nth partial sum satisfies ∣Rnβˆ£β‰€an+1=∣un+1∣|R_n|\leq a_{n+1}=|u_{n+1}| and has the same sign as the first omitted term un+1u_{n+1}.

  • The sum SS of a convergent alternating series lies between consecutive partial sums: S2n+1≀S≀S2nS_{2n+1}\leq S\leq S_{2n}.

  • For un=(βˆ’1)nn+1u_n=\frac{(-1)^n}{n+1}, the magnitudes an=1n+1a_n=\frac1{n+1} decrease to zero, so the series converges by the Alternating Series Test.

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Practice quiz preview

1. For the sequence un=(βˆ’1)n(n+2)n+1u_n=\frac{(-1)^n(n+2)}{n+1}, which sequence gives the magnitudes in its alternating-series representation?

2. A real sequence has terms u0=5u_0=5, u1=βˆ’3u_1=-3, and u2=2u_2=2. What sign must u3u_3 have if the sequence is alternating?

3. A series has terms un=(βˆ’1)nanu_n=(-1)^n a_n, where anβ‰₯0a_n\geq 0 decreases for all sufficiently large nn and tends to zero. What conclusion does the Alternating Series Test give?

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Flashcard preview

How are the terms of a real alternating series expressed using a positive sequence?

un=(βˆ’1)nanu_n=(-1)^n a_n or un=(βˆ’1)n+1anu_n=(-1)^{n+1}a_n, where an=∣un∣>0a_n=|u_n|>0.

What conditions ensure convergence under the Alternating Series Test for un=(βˆ’1)nanu_n=(-1)^n a_n?

anβ‰₯0a_n\geq 0 is eventually decreasing and tends to zero.

When does the alternating Riemann series converge?

It converges if and only if Ξ±>0\alpha>0.

When is a numerical series βˆ‘un\sum u_n absolutely convergent?

When βˆ‘βˆ£un∣\sum |u_n| converges.

What does convergence of βˆ‘βˆ£un∣\sum |u_n| imply about βˆ‘un\sum u_n?

βˆ‘un\sum u_n converges.

What bound holds when βˆ‘βˆ£un∣\sum |u_n| converges?

βˆ£βˆ‘n=0∞unβˆ£β‰€βˆ‘n=0∞∣un∣\left|\sum_{n=0}^{\infty}u_n\right|\leq\sum_{n=0}^{\infty}|u_n|.

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Frequently asked questions

What does the Series with Arbitrary Signs study guide cover?

It covers the core concepts of Series with Arbitrary Signs, organized by topic with the key definitions, explanations and takeaways, so it's easier to learn and remember.

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How many questions are in the Series with Arbitrary Signs quiz?

The practice quiz has 12 multiple-choice questions, each with an explanation. Use it to check what you know and find your gaps.

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How do I study Series with Arbitrary Signs with flashcards?

There are 29 flashcards on Series with Arbitrary Signs, with a question on the front and the answer on the back. Try to recall each answer before you flip, and let spaced repetition bring back the ones you miss.

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