Examples

p-series examples

By finding and applying the -series convergence properties:

We see that converges: so

But diverges: so

Integral test - pushing the envelope of convergence

Does converge?

Does converge?

Notice that grows very slowly with , so is just a little smaller than for large , and similarly is just a little smaller still.

Solution

(1) The two series lead to the two functions and .

Check applicability.

Clearly and are both continuous, positive, decreasing functions on .

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(2) Apply the integral test to .

Integrate :

Conclude: diverges.

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(3) Apply the integral test to .

Integrate :

Conclude: converges.

Direct comparison test: rational functions

(a)

Choose: and

Check:

Observe: is a convergent geometric series

Therefore: converges by the DCT.

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(b)

Choose: and .

Check:

Observe: is a convergent -series

Therefore: converges by the DCT.

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(c)

Choose: and

Check: (notice that )

Observe: is a convergent -series

Therefore: converges by the DCT.

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(d)

Choose: and

Check:

Observe: is a divergent -series

Therefore: diverges by the DCT.

Limit comparison test examples

(a)

Choose: and .

Compare in the limit:

Observe: is a convergent geometric series

Therefore: converges by the LCT.

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(b)

Choose: ,

Compare in the limit:

Observe: is a divergent -series

Therefore: diverges by the LCT.

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(c)

Choose: and

Compare in the limit:

Observe: is a converging -series

Therefore: converges by the LCT.