Series practice list handout - solutions

Main list

(1)

Use ratio test to determine convergence.

Since , the series absolutely converges.

Note: Many other methods work. E.g., can also do LCT with .

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(2)

Simplify series.

Consider absolute values:

This diverges ().

Now apply the AST. It is obvious that:

• Decreasing terms
• Limit to zero

So the AST says it converges. Therefore it converges conditionally.

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(3)

Start with DCT:

Now we guess that will diverge because it looks like a series.

Assuming this fact, the original series will diverge by the DCT.

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To show the fact, apply LCT with :

Since , and diverges, the LCT says that diverges also.

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(4)

This fails the simple divergence test:

Therefore the series diverges.

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(5)

Fails the simple divergence test:

Therefore the series diverges.

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(6)

Fails the simple divergence test:

Therefore the series diverges.

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(7)

Apply the ratio test:

By the ratio test, the series converges (absolutely).

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(8)

First check for absolute convergence:

Now move on to the AST:

The terms are decreasing and limit to 0. Therefore the AST applies and says that the series converges.

Since the associated positive series diverges, the series converges conditionally.

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(9)

Apply the root test:

By the root test, the series converges absolutely.

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(10)

Apply root test:

By the root test, the series converges absolutely.

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(11)

First test absolute convergence. The associated positive series is:

This series will converge but we need a comparison test, and we’ll use the LCT.

Let . Then:

Therefore, since , we know that and both converge or both diverge.

Since converges , we conclude that the original series converges absolutely.

At this point we are done, we do not need to consider the AST even though the series is alternating!

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(12)

First check for absolute convergence:

This thing will not converge. Compare to :

But diverges ().

So by the DCT, we know also diverges.

Now, is the original alternating series converging?

• Terms decreasing?

Yes. This is pretty obvious. You can also take the derivative to verify:

This is negative when .

• Terms limit to zero?

Yes. Also pretty obvious, considering but . To prove it more carefully, use L’Hopital’s Rule:

Now we know the AST applies and says that the series converges.

Since the associated positive series diverges, the series converges conditionally.

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(13)

Use the integral test to determine convergence.

Since the improper integral converges, by the integral test we know that the series converges.

Extra - Rogawski problems - Hints

(43) Geometric series. (Split the sum.)

(44) Ratio test.

(45) Ratio test.

(46) Integral test.

(47) LCT (easier) or IT (harder), not DCT

(48) DCT.

(49) It’s a -series.

(50) SDT

(51) Geometric series.

(52) -series (associated positive series) and AST

(53) LCT

(54) SDT

(55) SDT

(56) Root test