Series Slides A review of convergence tests Roxanne M. Byrne University of Colorado at Denver.

Series Slides

A review of convergence tests

Roxanne M. ByrneUniversity of Colorado at Denver

Nth Term Test

This test can be applied to any series

Nth Term Test

You must evaluate:

Lim an n

Where { a n } is the sequence of terms of the series

Nth Term Test

Conclusion:

n If lim a n 0, the series diverges

If lim a n = 0, the test failsn


Nth Term Test

Remarks:• Remember, if the limit is zero,

THE TEST FAILS. This means you must try a different test.

• Sometimes the limit is not easy to evaluate. In this case, try other test that you think might be more productive first.

• Conversely, some of the other tests need thislimit evaluated also. Remember this test if the limit is not zero.

Integral Test

This test can be applied only to positive term series

INTEGRAL TEST

You must:

• Find a continuous function, f(x), such that f(n) = an

• Verify that f(x) is a decreasing function

• Determine if f(x) dx converges


INTEGRAL TEST

Conclusion:

If the integral converges

then the series converges

If the integral diverges

then the series diverges

INTEGRAL TEST

Remarks:

• This is both a convergence and divergence test

• If f(x) is an increasing function, go to the

Nth Term Test.

• This test requires that the function can be

integrated. It will not work for series whose

terms have factorials in them.

Comparison Test


COMPARISON TEST

You must:

• Decide if you think the series converges or

diverges

• If you think it converges, you must find a

larger termed series that you know

converges.

• If you think it diverges, you must find a

smaller positive termed series that you

know diverges

COMPARISON TEST

Conclusion:

• If you find a larger termed convergent series,

then your series converges.

• If you find a smaller positive termed divergent

series, then your series diverges.

• If you cannot find an appropriate comparison

series, the test fails.

COMPARISON TEST

Remarks:• As with the Nth Term Test, when the test fails,

it means you must try another test.

• The test works well with series that look almost like a geometric series or a p-series.

• The major disadvantages of this test: You must decide beforehand if the series converges or diverges. You must find a corresponding

comparison series

Limit Comparison Test


LIMIT COMPARISON TEST

You must:• Decide if you think the series converges or

diverges

• If you think it converges, you must find a positive termed convergent series thathas the same end behavior as yours.

• If you think it diverges, you must find a positive termed divergent series thathas the same end behavior as yours.

• Evaluate where an and bn are

the terms of your two series

n

n

n a

blim

• If 0 < < , then both series

converge or both series

diverge.

• If equals zero or increases

without bound or does not exist, then

test fails.


n

n

n a

blim

Conclusion:

n

n

n a

blim


• When the test fails, you must either find anothercomparison series or you must try another test.

• The test works well with series that look almostlike a geometric series or p-series.

• The major disadvantages of this test: You must decide beforehand if the series

converges or diverges. You must find a corresponding

comparison series

Remarks:

Ratio Test


You must:

• Evaluate u n + 1

• Evaluate the ratio

• Evaluate lim

RATIO TEST

n

n

u

u 1

n

n

u

u 1

n

Where { u n } is the sequence of terms

RATIO TEST

Conclusion:

If the limit < 1 then

the series converges

If the limit > 1 then

the series diverges

If the limit = 1 then

the test fails

Remarks:

• This is both a convergence and divergence test

• This test can be used to prove absolute

convergence

• This test will not work on series whose terms

are rational functions of n. For these

series, use the Limit Comparison Test

and the end behavior of the terms.

RATIO TEST

• This test works well with series whose terms have factorials in them.

The N th Root Test


THE NTH ROOT TEST

You must:

• Find

• Evaluate

nna

nn

nalim


THE NTH ROOT TEST

Conclusion:

If the limit < 1 then

the series converges

If the limit > 1 then

the series diverges

If the limit = 1 then

the test fails

THE NTH ROOT TEST

Remarks:• This is both a convergence and divergence test

• This test can be used to prove absolute

convergence • This test will not work on series whose terms are

rational functions of n. For these series,

use the Limit Comparison Test and the end

behavior of the terms.

• This test works well with series whose terms have powers of n in them.

• This test does not work well with series whoseterms have factorials in them.

Absolute Convergence Test

This test is used on series with varying signed terms

ABSOLUTE CONVERGENCE TEST

You must:

• Let bn be the absolute value of the

sequence of terms of your series

• Determine if the sum of bn is a convergent

series by one of the positive term

convergence tests.

Conclusion:

• If the sum of bn converges then the

original series converges absolutely

• If the sum of bn converges then the

original series converges conditionally

or it diverges.


Remarks:

• If the sum of bn diverges then you usually

use the alternating series test to

determine if the original series converges.

• If you want to determine the type of

convergence of an alternating series,

you would use this test first.


Alternating Series Test

This test can be applied only to series that have

alternating terms

You must:

• Make sure the terms are alternating

• Define a new sequence, un, as the absolute value of the terms of your sequenceof terms.

• Prove that un is a decreasing sequence.

• Evaluate lim un

ALTERNATING SERIES TEST

n


Conclusion:

If the limit is zero,

then alternating series

converges.


• If you need to determine if the series is absolutely

or conditionally convergent, you must test to

see if un converges using a positive term

series test.

• If the lim un 0 or if un is an increasing

sequence, use the N th Term Test.

Remarks:

The EndThe End

PLEASE CLOSE WINDOW

Series Slides A review of convergence tests Roxanne M. Byrne University of Colorado at Denver.

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