11.1 SEQUENCES

Spring 2010Math 2644

Ayona Chatterjee

• A sequence can be thought of as a list of numbers written in a definite order.

• Here a1 is called the first term, a2 is the second term and an is the nth term.

• We will deal with only infinite sequences.

Examples of sequences

Theorems to keep in mind

SQUEEZE THEROREM

11.2 SERIES

11.3 THE INTEGRAL TEST

Introduction

• It is difficult to find the exact sum of series.• Easy only in case of geometric series.• We develop tests that enable us to determine

if the series converges or diverges without explicitly finding the sum.

Examples

• Let us look at a series whose terms are reciprocals of the squares of the positive integers.

• It appears as n goes to infinity that series is convergent.

• If we exclude the first rectangle we have

• This is an improper integral and we have shown it converges to 1.

• Thus we can say that

1

2 )/1( dxx

NOTE

11.4 THE COMPARISON TEST

Hints

• Constant terms in the denominator can usually be deleted without affecting the convergence or divergence of the series.

• If a polynomial in ‘k’ appears as a factor in the numerator/denominator, all but the leading term in the polynomial can usually be discarded without affecting the convergence or divergence of the series.

11.5 ALTERNATING SERIES

11.6 ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS

Strategy for testing series

• Check if it is a p-series or a geometric series and use standard known results.

• If the series is similar to a geometric series or a p-series, use the Comparison test or the Limit comparison test.

• Use the Test of divergence, if at a glance you can observe that

.0lim n

na

Strategies continued

• If an = f(n) where is easily evaluated, then use the Integral Test. (provided all conditions are satisfied.)

• If the series if of the form then use the Alternating series test.

• Series that involve factorials or other products (including a constant raised to the nth power) can be conveniently tested using the Ratio Test.

dxxf

1)(

)1(or )1( 1n

nn

n bb

Strategies continued

• Note that for all p-series and therefore for all rational and algebraic functions of n. Thus the Ratio Test should not be used for such series.

• If an is of the form then the Root test may be useful.

naa nn as 1/1

nnb )(

11.8 POWER SERIES