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W08 - HW Solutions

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03 - Limit Comparison Test

Use the Limit Comparison Test to determine whether the series converges:

Show your work. You must check that the test is applicable.

Solution

Find appropriate for comparison.

A good would be .

Verify applicability of the limit comparison test.

since for all and for all .

since for all .

Apply limit comparison test.

Interpret results.

Since and diverges via the -series test, we conclude that also diverges.

05 - Integral Test

Determine whether the series is convergent by using the Integral Test. Show your work. You must check that the test is applicable.

(a) (b) (c)

Solution

(a)

Verify applicability of the integral test.

since for all .

is continuous for all . (only discontinuity is at , but the series starts at ).

is monotone decreasing, since as increases, the denominator increases, and the term decreases.

Apply the integral test.

Interpret results.

Since converges, so must .

(b)

Verify applicability of the integral test.

since and for all .

is continuous for all .

has critical points at . When , , so is monotone decreasing.

Apply the integral test.

Interpret results.

Since converges, so must .

(c)

Verify applicability of the integral test.

since for all .

is continuous for all . (The only discontinuity is at , but the series starts at .)

is monotone decreasing, since as increases, the denominator increases, and the term decreases.

Apply the integral test.

Interpret results.

Since converges, so must .

06 - Integral Test, Direct Comparison, Limit Comparison

Determine whether the series converges by checking applicability and then applying the designated convergence test.

(a) Integral Test:

(b) Direct Comparison Test:

(c) Limit Comparison Test:

Solution

(a)

Verify applicability of the integral test.

since for all , and for all .

is only discontinuous at , but since the series starts at , this discontinuity is not relevant.

. The critical point is at . For , , so the function is decreasing.

Apply the integral test.

Derive conclusions.

Since converges, so must .

(b)

Find appropriate for comparison.

A good would be .

Verify applicability of the direct comparison test.

Since for large because the denominator for is always larger, the DCT can be applied.

Apply the direct comparison test.

Since converges by the -series test, we conclude that must converge as well.

(c)

Find appropriate for comparison.

A good would be .

Verify applicability of the limit comparison test.

since the numerator is always positive and for all .

is always positive since for all .

Apply limit comparison test.

Interpret results.

Since and converges via the -series test, we conclude that also converges.

07 - Limit Comparison Test

Use the Limit Comparison Test to determine whether the series converges:

Show your work. You must check that the test is applicable.

Solution

Find appropriate for comparison.

A good would be .

Verify applicability of the limit comparison test.

Since for all (note that grows faster than ) and for all , the test is applicable.

Apply the limit comparison test.

Determine convergence of using the integral test.

Derive conclusions about and .

Since converges by the integral test and , we conclude that also converges.