TESTING INFINTE SERIES FOR CONVERGENCE-2 |

**TABLE OF CONTENTS**

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With both the basic and limit comparison tests, **we chose ****, a series we compare** with -- the one **we're testing**. The basic test sometimes fails because we're unable to specify that the terms of are always less/greater than or equal to the terms in .That's when we use the limit comparison test.

**Basic Comparison Test**:

1) if , the **comparison** series, **converges;** and the terms of , the **test** series, are **smaller** **than** or equal to the terms of the **comparison series**, the **test** series also **converges**.

2) if the **comparison** series **diverges** and the terms of the **test** series are **greater** **than** or equal to the terms of the **comparison series**, the **test** series also **diverges**.

More formally:

1) If is a convergent series and for all values of n, then is also convergent.2) If is a |

The comparison series is often a **geometric** or ** p-series**.

**Example**

Test for convergence.

For , we'll choose the **convergent p-series**

Now, since n^{5} + 4, (the denominator of the test series) is greater than n^{5}, (the denominator of the comparison series), the terms of are always smaller than those of , because we're dividing by a larger value. This means that is also convergent.

**Example**

Test for convergence.

For , we'll choose the **divergent p-series** . Problem is, when we choose a divergent comparison series, we have to state that , to show divergence of .

Here however, , so we

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In a case like the last example, when we can't state that the right inequality relation holds between the two series to satisfy the conditons of the Basic Test, we use the Limit Test.

**The Limit Comparison Test says**:

We take the **limit at infinity** of the test series divided by the comparison series .

If **we get a positive constant or infinity**, the two **series are the same**.

That is, if converges, converges and if S *b*_{ n} diverges, diverges.

More formally:

If , then both series converge or both series diverge. |

Let's finish the last example.

We take the limit at infinity of the test series divided by the comparison series.

Since both numerator and denominator have *n*^{2 / 3} , the limit equals 1.

Therefore, the series is divergent since the limit is positive.

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**Choosing the Comparison Series **

When we need a comparison series to test a fraction, we **choose the terms** in both numerator and denominator **that have the greatest effect on the magnitude**. Since we're taking limit at infinity, if we have 3n² + 5, it's pretty obvious that the 5 has little effect on the value of the expression once *n* approaches infinity.

**Example:** Test for convergence.

Once we approach infinity, the 5*n* will have little effect on the numerator because 3*n*² grows large faster than 5*n* does. Same goes for the 1 in the denominator.

So, our comparison series in this instance will be

It is a **convergent G**eometric **S**eries with *a* = 3/2 and *r* = ½ .

Now we take the limit .

So the test series is also convergent.

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**memory tweak:** to find limit at infinity of a fraction, divide every term in the fraction by the highest power of the variable in the fraction, then limit the variable to . In the example above, we divided top and bottom by *n*², cancelled the 3's and that's how we got limit = 1.

**Example:** Test for convergence.

Our comparison series in this instance will be

This is a ** p-series **with

Now we take the limit .

So the test series is also convergent.

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With this test, we take the limit at infinity of the absolute value of the (*n* + 1)*th* term divided by the *n*th term. If that limit is smaller than 1, the test series converges. If it is bigger than 1, the test series diverges. If the limit equals 1, we need to use another test.

If , the series converges if L < 1the series diverges if L > 1.The test is inconclusive when L = 1. |

If the series includes only positive terms, we can ignore the absolute value signs.

Useful if ** a_{n}** involves factorials or

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**memory tweak:** Remember that *n*! = 1 % 2 % 3 % 4 % .......... % *n*.

When we divide ( *n* + 1)! by *n*! we're left with *n* + 1.

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**Example:** Use the Ratio Test to test the convergence of .

The limit is

Since 0 < 1, the series **converges**.

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**Example:** Use the Ratio Test to test the convergence of .

The limit is

Since 3 > 1, the series **diverges**.

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With this test, we take the limit at infinity of the absolute value of the *n*th root of *a*_{ n}. Then the conditions are the same as for the Ratio Test. Remember that .

If , the series converges if L < 1the series diverges if L > 1.The test is inconclusive when L = 1. |

If the series includes only positive terms, we can ignore the absolute value signs.

Useful if ** a_{n}** involves only

**Example:** Use the Root Test to test the convergence of .

The limit is

- Since 0 < 1, the series

Test these infinite series for convergence.

State the test you use and if the series converges or diverges.

1) | 2) |

3) | 4) |

5) | 6) |

7) | 8) |

9) | 10) |

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1) Basic Comparison with b_{n} = 1/n^{4}.convergent series converges. |
2) Basic Comparison with b_{n} = 3n/2n² = 3/2n
series diverges. |

3) Limit Comparison test b_{n} = 1/n^{ 3 / 2}.convergent series converges. |
4) Limit Comparison test b_{n} = 1/e^{ n}.convergent Geometric series series converges. |

5) Basic Comparison with b_{n} = 1/2^{ n}.since | sin convergent Geometric series series converges. |
6) Ratio Test
Since 3 > 1, series diverges. |

7) Ratio Test series diverges. |
8) Ratio Test
series converges. |

9) Root Test
series diverges. |
10) Root Test series converges. |

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