types of series and tests for convergences

Chapter 9 Infinite Series

MATH- 102Calculus-II

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Series

When the terms of a sequence are added, we get a series

...,25,16,9,4,1The sequence

gives the series ...2516941

Sigma Notation for a Series

A series can be described using the general term

100...2516941 e.g.

10

1

2ncan be written

is the Greek capital letter S, used for Sum

1st value of n

last value of n

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

9.3 INFINITE SERIES (pg:614)

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Definition 9.3.1 (p. 614) INFINITE SERIES

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Definition 9.3.2 (p. 616)

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Theorem 9.3.3 (p. 617)

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Figure 9.3.4 (p. 620)

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

9.4 Convergence Tests (pg: 623)

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Theorem 9.4.1 (p. 624)The Divergence Test

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Theorem 9.4.2 (p. 624)

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Theorem 9.4.3 (p. 625)

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Theorem 9.4.4 (p. 626)The Integral Test

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Figure 9.4.1 (p. 626)

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Theorem 9.4.5 (p. 629)Convergence of p-Series

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

9.5 The Comparison, Ratio, and Root Tests (pg: 629)

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Theorem 9.5.1 (p. 631)The Comparison Test

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Figure 9.5.1 (p. 632)

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Theorem 9.5.4 (p. 633)The Limit Comparison Test

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Theorem 9.5.5 (p. 634)The Ratio Test

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Theorem 9.5.6 (p. 635)The Root Test

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Equations (1) and (2) (p. 638)

Alternating Series

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

9.6 Alternating Series (pg: 638)

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Definition 9.6.3 (p. 641)

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Theorem 9.6.4 (p. 642)

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Theorem 9.6.5 (p. 644)Ratio Test for Absolute Convergence

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Page 645Summary of Convergence Tests

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Page 645 Summary of Convergence Tests

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Definition 9.7.2 (p. 650)

Definition 9.7.3 (p. 653)

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Figure 9.7.3 (p. 654)