Problems

01

Absolute and conditional convergence

Determine whether the series are absolutely convergent, conditionally convergent, or divergent.

Show your work. You must check applicability of tests.

(a) (b)

Solution

03

(a)

Therefore it is not absolutely converging. Proceed to the AST.

• Passes SDT? Yes, .
• Decreasing? Yes, denominator is increasing.

Therefore the AST applies and says this converges. So it converges conditionally.

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(b)

So this fails the SDT, hence it diverges.

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02

Absolute and conditional convergence

Determine whether the series are absolutely convergent, conditionally convergent, or divergent by applying series tests.

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

(a) (b)

Solution

07

(a) We first check for absolute convergence:

This fails the SDT, so the series diverges!

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(b) Notice that . Check for absolute convergence:

Then:

Since converges (), the DCT says that converges. So the original series converges absolutely and we are done.

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03

Alternating series: error estimation

Find the approximate value of such that the error satisfies .

How many terms do you really need?

Solution

08

We use the alternating series test error bound formula. (AKA: “Next Term Bound”)

We seek the smallest such that . What that happens, we will have:

Our formula for :

We cannot easily solve for to provide , so we just start listing out the terms:

We see that is the first term less than 0.005, so and we need the first 5 terms.

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