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W10 - HW Solutions

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01 - Ratio and root tests

Apply the ratio test or the root test to determine whether each of the following series are absolutely convergent, conditionally convergent, or divergent.

(a) (b) (c)

Solution

(a) Apply the root test.

Compute limit

State conclusions.

Since the limit is less than 1, we conclude that the series converges absolutely.

(b)

Apply the ratio test.

Compute limit.

State conclusions.

Since the limit is less than 1, we conclude that the series is absolutely convergent.

(c)

Apply the ratio test.

Compute limit.

State conclusions.

Since the limit is less than 1, we conclude that the series is absolutely convergent.

03 - Ratio and root tests

Apply the ratio test or the root test to determine whether each of the following series is absolutely convergent, conditionally convergent, or divergent.

(a) (b) (c)

Solution

(a)

Apply ratio test.

Compute limit.

State conclusions.

Since the limit is greater than 1, we conclude that the series is divergent.

(b)

Apply root test.

Compute limit.

State conclusions.

Since the limit is less than 1, we conclude that the series is absolutely convergent.

(c)

Apply ratio test.

Compute limit.

State conclusions.

Since the limit is 1, we conclude that the series is absolutely convergent.

04 - Various limits, Part I

Find the limits. You may use or or as appropriate. Braces indicate sequences.

  • C = Convergent
  • AC = Absolutely Convergent
  • CC = Conditionally Convergent
  • D = Divergent

C or D
C or D
AC, CC, or D
AC, CC, or D
00
1
00
00
00

05 - Various limits, Part II

Find the limits. You may use or or as appropriate. Braces indicate sequences.

  • C = Convergent
  • AC = Absolutely Convergent
  • CC = Conditionally Convergent
  • D = Divergent

C or D
C or D
AC, CC, or D
AC, CC, or D
00
00
00

07 - Power series - interval of convergence

Find the radius and interval of convergence for these power series:

  • (a)
  • (b)

Solution

(a)

Apply the ratio test.

Find interval of convergence.

Check endpoints and state final answer

at , we have , which converges absolutely by the alternating series test.

at , we have , which converges by the DCT.

Therefore, the radius of convergence is and our interval of convergence is .

(b)

Apply the ratio test.

Derive conclusions.

Since the ratio test evaluates to 0, the series converges for all , so the radius of convergence is and the interval of convergence is .