SERIES

 

Known Series    

Tests for Convergence

Divergence Test      Integral Test      Comparison Test      Ratio Test      Root Test

Limit Comparison Test        Alternating Series Test      Ratio Test for Absolute Convergence

 

 

Convergence Tests

Divergence Test      Integral Test      Comparison Test      Ratio Test      Root Test

Limit Comparison Test        Alternating Series Test      Ratio Test for Absolute Convergence

 

 NAME 

STATEMENT

COMMENTS

Divergence Test

If lim uk ¹0, then Suk diverges.

 

If lim uk = 0, Suk may or may not converge.

Divergence Test      Integral Test      Comparison Test      Ratio Test      Root Test

Limit Comparison Test        Alternating Series Test      Ratio Test for Absolute Convergence

 
 

 NAME 

STATEMENT

COMMENTS

Integral Test

Let Suk be a series with positive terms, and let f(x) be the function that results when k is replaced by x in the formula for uk. If f is decreasing and continuous for x ³ 1, then

¥                           + ¥
åuk    and     ò1        f(x) dx
k = 1
both convergence or both divergence.  

 

Use this test when f(x) is easy to integrate.

 

This test only applies to series that have positive terms.

Divergence Test      Integral Test      Comparison Test      Ratio Test      Root Test

Limit Comparison Test        Alternating Series Test      Ratio Test for Absolute Convergence

 

 NAME 

STATEMENT

COMMENTS

Comparison Test

Let åak and åbk be series with nonnegative terms such that

       a1 £ b1, a2 £ b2 ,…, ak £ bk ,….

If åbk  converges, then åak  converges, and if åak diverges, then åbk diverges.

 

Use this test as a last resort; other tests are often easier to apply.

This test only applies to series with nonnegative terms.

Divergence Test      Integral Test      Comparison Test      Ratio Test      Root Test

Limit Comparison Test        Alternating Series Test      Ratio Test for Absolute Convergence

 

 NAME 

STATEMENT

COMMENTS

Ratio Test

Let åuk be a series with positive terms and suppose

                   uk +1
         lim     ¾¾ = r
            k®+¥     uk

(a)        Series converges if r < 1.

(b)        Series diverges if r > 1 or r = + ¥.

(c)        No conclusion if r = 1.

Try this test when uk involves factorials or kth powers.

Divergence Test      Integral Test      Comparison Test      Ratio Test      Root Test

Limit Comparison Test        Alternating Series Test      Ratio Test for Absolute Convergence

 

 NAME 

STATEMENT

COMMENTS

Root Test

Let åuk be a series with positive terms such that   r = lim  
          k®+¥

(a)      Series converges if r < 1.

(b)      Series diverges if r > 1 or r = + ¥.

(c)      No conclusion if r = 1.

Try this test when uk involves kth powers.

Divergence Test      Integral Test      Comparison Test      Ratio Test      Root Test

Limit Comparison Test        Alternating Series Test      Ratio Test for Absolute Convergence

 

 NAME 

STATEMENT

COMMENTS

Limit Comparison Test

Let åak and åbk be series with positive terms such that

                  ak
r = lim      ¾
     k®+¥      bk

If 0 < r < + ¥, then both series converge or both diverge.

 

 This is easier to apply than the comparison test, but still requires some skill in choosing the series åbk for comparison.

Divergence Test      Integral Test      Comparison Test      Ratio Test      Root Test

Limit Comparison Test        Alternating Series Test      Ratio Test for Absolute Convergence

 

 NAME 

STATEMENT

COMMENTS

Alternating Series Test

The series
a1a2 + a3a4 +…
           and
-a1 + a2a3 + a4 -…
converge if

(a)      a1 > a2 > a3 ….

(b)      lim  ak = 0
k®+¥

 

This test applies only to alternating series.

It is assumed that ak > 0 for all k.

Divergence Test      Integral Test      Comparison Test      Ratio Test      Root Test

Limit Comparison Test        Alternating Series Test      Ratio Test for Absolute Convergence

 

 NAME 

STATEMENT

COMMENTS

Ratio Test for Absolute Convergence

Let åuk  be a series with nonzero terms such that:

       (a)      Series converges absolutely if  r <1
(b)      Series diverges if r > 1 or r = + ¥
(c)      No conclusion if r = 1

 

 

The series need not have positive terms and need not be alternating to use this test.

Note about absolute convergence:

If 

converges, then so does  

That is, every absolutely convergent series converges.    

Divergence Test      Integral Test      Comparison Test      Ratio Test      Root Test

Limit Comparison Test        Alternating Series Test      Ratio Test for Absolute Convergence