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 u_k ¹0, then Su_k diverges.

If lim u_k = 0, Su_k may or may not converge.

 NAME 

STATEMENT

COMMENTS

Integral Test

Let Su_k 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 u_k. If f is decreasing and continuous for x ³ 1, then

¥                           + ¥
åu_k    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.

 NAME 

STATEMENT

COMMENTS

Comparison Test

Let åa_k and åb_k be series with nonnegative terms such that

       a₁ £ b_1, a₂ £ b₂ ,…, a_k £ b_k ,….

If åb_k  converges, then åa_k  converges, and if åa_k diverges, then åb_k diverges.

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

This test only applies to series with nonnegative terms.

 NAME 

STATEMENT

COMMENTS

Ratio Test

Let åu_k be a series with positive terms and suppose

                   u_{k +1}
         lim     ¾¾ = r
            k®+¥     u_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 u_k involves factorials or kth powers.

 NAME 

STATEMENT

COMMENTS

Root Test

Let åu_k 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 u_k involves kth powers.

 NAME 

STATEMENT

COMMENTS

Limit Comparison Test

Let åa_k and åb_k be series with positive terms such that

                  a_k
r = lim      ¾
     k®+¥      b_k

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 åb_k for comparison.

 NAME 

STATEMENT

COMMENTS

Alternating Series Test

The series
a₁ – a₂ + a₃ – a₄ +…
           and
-a₁ + a₂ – a₃ + a₄ -…
converge if

(a)      a₁ > a₂ > a₃ ….

(b)      lim  a_k = 0
k®+¥

This test applies only to alternating series.

It is assumed that a_k > 0 for all k.

 NAME 

STATEMENT

COMMENTS

Ratio Test for Absolute Convergence

Let åu_k  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 

That is, every absolutely convergent series converges.