Calculus II
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W08 - HW Solutions

01

First note that:

  1. is continuous
  2. is positive
  3. is monotone decreasing because is increasing

Then:

Since this is finite, the integral test establishes that the series converges.

02

(a)

First term:

Common ratio is .

Geometric series summation formula, always first term on top:

(b)

First term:

Common ratio:

Geometric series summation formula:

03

(a) diverges (b) (c) diverges (d)

(e)

(f) diverges:

(g)

(h)

Use L’Hopital’s Rule:

04

(a)

(b) diverges:

(Note that .)

(c) :

L’Hopital’s Rule:

(d) by (c), the sign doesn’t affect convergence to

(e) :

Multiply above and below by :

(f) :

This is a well-known formula for . If that formula is not used as the definition of , then it would not be circular reasoning to argue as follows:

(g) diverges:

So:

05

(a)

First term: .

Common ratio:

Geometric series summation formula:

(b)

This is geometric starting with .

First term: .

Common ratio: .

Geometric series summation formula:

Add back the first term:

06

Compute the first few areas, with being the area of the largest triangle:

This is a geometric series with :

Geometric series total sum formula:

07

(a)

The first term is . The common ratio is . Therefore the sum:

(b)

Split numerator and obtain two geometric series:

Geometric series total sum formula:

08

(a) Verify applicability of the integral test:

  • is continuous for all . (Only discontinuity is at , but the series starts at .)
  • since for all .
  • is monotone decreasing, since as increases, the denominator increases, and the term decreases.

Apply the integral test:

This is finite and the improper integral converges, so the series converges by the Integral Test.

(b) Verify applicability of the integral test with :

  • is definitely continuous for all .
  • since and for all .
  • Decreasing?
    • has zeros at .
    • When , .
    • Series starts at , so the terms are monotone decreasing.

Apply the integral test:

This is finite and the improper integral converges, so the series converges by the Integral Test.

(c) Verify applicability of the integral test for :

  • is continuous for all . (The only discontinuity is at , but the series starts at .)
  • since for all .
  • is monotone decreasing, since as increases, the denominator increases, and the term decreases.

Apply the integral test:

This is finite and the improper integral converges, so the series converges by the Integral Test.