Geometric Series Test. Basic Divergence Test. P-Series .

Geometric Series Test.Basic Divergence Test.p-Series Test.Integral Test.Basic Comparison Test.Limit Comparison Test.Root TestRatio Test

Alternating Series Test for Convergence: ( 1)If the alternating series n 1n 1bn b1 b2 b3 b 4 b5 b6 . bn 0satisfies(a) bn 1 bnthen the series is convergent.for all nbn 0(b) nlim bn 0 , the series diverges by the Basic Test forNote: If nlim Divergence, NOT by the Alternating Series Test.Ex. n 1( 1)2nn

Absolute and Conditional ConvergenceIf the series an 1 an 1n 1converges, then the series an 1nalso converges. n an nis absolutely convergent if a n converges.n 1is conditionally convergent if an 1nconverges but an 1ndiverges.Note (rearrangement invariance):If a series converges absolutely, then it will converge to the same valueregardless of the order in which the terms are summed.If the series converges conditionally, then the terms of the series can berearranged to sum to any desired value.

Popper1. n 1( 1)n2na. converges absolutelyb. converges conditionallyc. diverges2. 2n 1 5n 3 2nn 1a. convergesb. diverges

Popper3. 3n 1 5n 2 2nn 1a. convergesb. diverges4. n 3nn 1a. convergesb. diverges

Popper5. n 1( 1)nn2a. converges absolutelyb. converges conditionallyc. diverges

Popper6. ( 1)n 1n 1arctan ( n )n2a. converges absolutelyb. converges conditionallyc. diverges

7. n 1n cos ( n π )2na. converges absolutelyb. converges conditionallyc. diverges8. n cos ( n π )n 1n2 1 a. converges absolutelyb. converges conditionallyc. diverges

Popper9. ln n nn 1a. convergesb. diverges

Alternating Series RemainderIf a convergent alternating series satisfies the condition 0 an 1 an, thenthe remainder R N involved in approximating the sum S by S N is less inmagnitude than the first neglected (truncated) term. That is,R N S SN a N 1.The alternating Series Remainder is called remainder, error or S SN .Ex: Approximate the sum ofthe error. n 1( 1)n 1 1 by its first six terms, and find n!

Ex: Approximate the sum of n 1( 1 )n 1n4with an error of less than 0.001.

Section 11.5Taylor Polynomials in x

There are many functions that we only know at one point, or a handful ofisolated points. Such as the trigonometric functions, ex, ln x, etc.Let’s create a polynomial P(x) that has the same properties as somefunction f (x) that we know very well at x a, such as sin (x) or ex aroundx 0.The properties that we need to consider are the function and derivativeproperties.Why a polynomial?

1) Find a polynomial of degree n 4 for f ( x ) e x . 3 2 1123

3 2 1123 3 2 1123 3 2 1123 3 2 1123

2) Find a polynomial of degree n 6 for f ( x ) cos x . 3 2 1123

3 2 1123 3 2 1123 3 2 1123 3 2 1123

3) Find a polynomial of degree n 5 for f ( x ) sin x . 3 2 1123

3 2 1123 3 2 1123 3 2 1123 3 2 1123

Definition of nth degree Taylor polynomial:If f has n derivatives at c, then the polynomial(n )f '' ( c )f(c )2nPn ( x ) f ( c ) f ' ( c )( x c ) ( x c ) . (x c)2!n!is called the nth degree Taylor polynomial for f at c.If c 0, thenPn ( x ) f ( 0 ) f ' ( 0 ) x f '' ( 0 )2x . f(n )(0 )2!n!may be called the nth degree Maclaurin polynomial for f.xn

x2 x34) Use the Taylor approximation e 1 x for x near 0 to find: 2! 3!ex 1.limx 0 2xx

Basic Divergence Test. p-Series Test. Integral Test. Basic Comparison Test. Limit Comparison Test. Root Test Ratio Test . Alternating Series Test for Conve