W07 - HW Solutions

01

(1) Indeterminate form:

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(2) L’Hopital:

Convert:

Change to and apply L’Hopital:

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(3) Take limit:

Therefore as .

02

(a) (b) diverges (c)

(d)

Observe that as , but for each , the value is below , in the domain of , which is continuous for .

(e)

03

04

(a)

(b)

(c)

(d)

(e)

(f)

05

(a)

(b)

(c)

06

(a)

(1) Set up squeeze relations:

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(2) Apply theorem:

We have:

Therefore:

We conclude that converges.

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(b)

(1) Generate squeeze inequalities:

Observe:

Rewrite RHS:

Raise all terms to :

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(2) Apply squeeze theorem:

Therefore:

Conclude that:

07

(a)

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(b)

When , this formula is undefined, because is undefined. But we know that:

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(c)

Simply take the limit of as :

08

(a)

(1) Recall geometric partial sum formula:

This one may be easiest to recall:

(Note here is the first term in the summation so it appears in the formula.)

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(2) Identify ingredients in partial sum formula:

Rewrite summand to determine and :

We see that and .

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(b) Take limit:

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Note A: The second term goes to zero: .

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(c)