Tests for Convergence of a Sequence:

Definition of Convergence of a Sequence:

Given any sequence {an}, the sequence is convergent if = L

Squeeze Theorem for Sequences

If an bn cn for all n N, and if = = L, then = L

Geometric Sequence: A geometric sequence { } is convergent if −1 < r 1 and divergent for all other values of r.

If −1 < r < 1, then

Tests for Convergence of a Series:

Definition of Convergence of a Series: Given any series , the series is convergent if the sequence

of partial sums {sn} is convergent to a number S; that is where

Evaluating the (Limit of) Partial Sum

If you are lucky and you find an explicit general formula the nth partial sum, expressed in

terms of n, then you can evaluate to see if it approaches a number S to find the sum of the series.

If the limit of the partial sum exists, , then the series is convergent with sum S.

If the limit of the partial sum does not exist, then the series is divergent.

Test for Divergence

Given any series , if or does not exist, then the series is divergent.

*** Test for divergence fail to conclude if

Special Series :

Geometric Series: is convergent for −1 < r < 1 and is divergent for r −1 or r 1.

*** If an infinite geometric series is convergent, then its sum is

p – Series: is convergent for p >1 and is divergent for p 1.

Integral Test

Suppose f(x) is a continuous, positive, decreasing function on the interval [1, ) and is a series

with an = f(n) for all integers n 1 :

then is convergent if and only if is convergent .

then is divergent if and only if is divergent .

Comparison Test

Suppose and are series with positive terms an > 0 and bn> 0 for all n 1 :

If is convergent and an bn for all n 1, then is also convergent.

If is divergent and an bn for all n 1, then is also divergent.

Limit Comparison Test

Suppose and are series with positive terms an > 0 and bn> 0 for all n 1 :

If for some finite number C > 0, C , then either both series converge or both

series diverge.

The comparison tests require that you have a known convergent or known divergent series that

you can compare on a term by term basis to the terms of

Ratio Test: Given a series ann1

:

If

< 1, then seriesan

n1

is absolutely convergent and therefore also convergent.

If

> 1 or = , then the series an

n1

is divergent.

If = 1 , the test gives no information about convergence or divergence of ann1

*** The ratio test is useful for series whose terms contain factorials.

Absolute-Convergence Test

Suppose an is an alternating series: If ann1

converges, then so does ann1

.

If ann1

converges, the series is "absolutely convergent" which is a stronger condition that

convergence. If you can find a test that works for series with positive terms to show that

ann1

converges, then this test tells you that ann1

also converges.

If ann1

diverges, then this test does not give you any information about the convergence or

divergence of the alternating series ann1

. You need to find another test of convergence.

*** An alternating convergent series an converges absolutely if an also converges

*** An alternating convergent series an converges conditionally if an diverges.

Alternating Series Test

Suppose is an alternating series , consider {bn} where bn = |an| :

The series converges if all the 3 conditions are satisfied.1) for all n 1

2)

3) bn+1 < bn for all n 1