TESTING INFINTE SERIES FOR CONVERGENCE-2

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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 divergent series and , for all values of n, then is also divergent.

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

Example

Test for convergence.

For , we'll choose the convergent p-series

Now, since n5 + 4, (the denominator of the test series) is greater than n5, (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 use the LIMIT comparison test to show is divergent.

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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 n2 / 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 5n will have little effect on the numerator because 3n² grows large faster than 5n does. Same goes for the 1 in the denominator.

So, our comparison series in this instance will be

It is a convergent Geometric Series 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 n², 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 p = 5/2 > 1 so it is convergent.

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 nth 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 < 1 the 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 an involves factorials or nth powers.

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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 nth root of a n. Then the conditions are the same as for the Ratio Test. Remember that .

 If , the series converges if L < 1 the 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 an involves only nth powers.

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

The limit is

Since 0 < 1, the series converges.

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 bn = 1/n4.convergent p-series. series converges. 2) Basic Comparison with bn = 3n/2n² = 3/2nbn = 3n/2n² or 3/2n, divergent harmonic series.series diverges. 3) Limit Comparison test bn = 1/n 3 / 2.convergent p-series.series converges. 4) Limit Comparison test bn = 1/e n.convergent Geometric series r = 1/e.series converges. 5) Basic Comparison with bn = 1/2 n.since | sin n | 1.convergent Geometric series r = 1/2.series converges. 6) Ratio Test Since 3 > 1, series diverges. 7) Ratio Testseries diverges. 8) Ratio Test series converges. 9) Root Test series diverges. 10) Root Testseries converges.

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