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Mathematical Expression Editor

Test 3 Review

1 Sample Test 3A

(problem 1) Determine whether the infinite series converges or diverges:

Use the Test for Divergence

Ratio of lead coeff:

(problem 2) Use the Integral Test to determine whether the infinite series converges
or diverges (if it applies): .

Show the derivative is negative for sufficiently large

Use u-sub to get the anti-derivative

Write the improper integral using a limit

The integral diverges

See problem 2a in section 3.5

(problem 3) Determine whether the infinite series converges or diverges:

Use the Direct Comparison Test

for all

Compare to -series with convergence

(problem 4) Determine whether the infinite series converges or diverges:

Use the
Limit Comparison Test

Compare with the divergent Harmonic Series

and

(problem 5) Determine whether the infinite series converges absolutely, converges
conditionally or diverges:

Use the Alternating Series Test to see that the series converges

Use the Limit Comparison Test (with the Harmonic Series) to see that the
non-alternating counterpart diverges

The alternating series converges conditionally

(problem 6) Determine whether the infinite series converges or diverges:

Use the Ratio Test

so the series converges absolutely (this is an alternating series)

(problem 7) Determine whether the infinite series converges or diverges:

Use the Root Test

Ratio of lead coeff: so the series converges

(problem 8) Find the interval of convergence of the power series (be sure to check
the endpoints):

Use the Ratio Test

The endpoints are and

The series diverges at both endpoints by the test for divergence

(problem 9) Find a power series representation for the function and include the
interval of convergence in your answer:

Factor from the numerator and from the denominator

Use the formula: with in place of

Answer:

2 Sample Test 3B

(problem 1) Determine whether the infinite series converges or diverges:

Use the Test for Divergence

(problem 2) Use the Integral Test to determine whether the infinite series converges
or diverges (if it applies): .

Show the derivative is negative for sufficiently
large

The anti-derivative is the inverse tangent function

Write the improper integral using a limit

The integral converges

See example 1 in section 3.5

(problem 3) Determine whether the infinite series converges or diverges:

Use the Limit Comparison Test with the Harmonic Series

(problem 4) Determine whether the infinite series converges or diverges:

Use the Direct Comparison Test

for all

Since , compare to -series with convergence

(problem 5) Determine whether the infinite series converges absolutely, converges
conditionally or diverges:

Use the Alternating Series Test to see that the series converges

Use the derivative to show that the terms are decreasing

Use the Limit Comparison Test (with the Harmonic Series) to see that the
non-alternating counterpart diverges

The alternating series converges conditionally

(problem 6) Determine whether the infinite series converges or diverges:

Use the Ratio Test

so the series converges

(problem 7) Determine whether the infinite series converges or diverges:

Use the Root Test

Ratio of lead coeff: so the series converges absolutely (this is an alternating
series)

(problem 8) Find the interval of convergence of the power series (be sure to check
the endpoints):

Use the Ratio Test

The endpoints are and

The series converges at (alternating Harmonic) and diverges at (Harmonic)

(problem 9) Find a power series representation for the function and include the
interval of convergence in your answer: