The Ratio Test Works Especially Well When the Problems Involve Factorials Or Exponential Powers

9.6--The Ratio Test (day #1)

The Ratio Test works especially well when the problems involve factorials or exponential powers.

Use the Ratio Test to determine the convergence or divergence of the series:

1) ∞ n ! 2n n=1 9.6--The Ratio Test (day #1)

Use the Ratio Test to determine the convergence or divergence of the series: ∞ 2) n+1 2 n 5 n=1

Use the Ratio Test to determine the convergence or divergence of the series:

3) ∞ (-1)n (3/4)n n3 n=1 9.6--The Ratio Test (day #1)

4) Verify that the ratio test is inconclusive here: ∞ 1 n3 n=1

The ratio test never works on a p-series!!! (The limit always as x approaches infinity is always 1.)

9.6--The Root Test (day #2)

This test is effective for series involving nth powers. 2n, 5n, 3n/nn, etc... 9.6--The Root Test (day #2)

Use the Root Test to determine the convergence/divergence of the series:

5) ∞ e2n nn n=1

Here's the same series done using the Ratio Test instead. Notice how the limit is tougher to evaluate this time: ∞ 5a) e2n nn n=1

e2(n+1) (n+1) 2n+2 n lim (n+1) lim e n (n+1) 2n n ∞ e2n n ∞ (n+1) e nn

2 n lim e n n ∞ (n+1)1 (n+1)n

e2 n n lim ( ( = 0 1 n ∞ (n+1) n+1 9.6--The Root Test (day #2)

Use the Root Test to determine the convergence/divergence of the series: ∞ 6) n n + 3 4n + 1 n=1

Use the Root Test to determine the convergence/divergence of the series: ∞ 7) 2n 4n 3n - 5 n=1 9.6--Review of Tests for Convergence (day #3)

So far, we've learned the following tests for the convergence/divergence of series:

Telescoping Series Test p-Series Test

Limit Comparison Test Alternating Series Test

Ratio Test Root Test

Determine the convergence or divergence of the series. Identify the test used: ∞ 51) (-1)n+1 5 n n=1

∞ 52) 100 n n=1 9.6--Review of Tests for Convergence (day #3)

Determine the convergence or divergence of the series. Identify the test used: ∞ 53) 3 n n n=1

∞ n 54) 2π ) 3 ) n=1

∞ 55) 5n 2n - 1 n=1

∞ 56) n 2n2 + 1 n=1 9.6--Review of Tests for Convergence (day #3)

Determine the convergence or divergence of the series. Identify the test used: ∞ 57) (-1)n 3n-2 2n n=1

∞ 58) 10 3 n3 n=1

∞ 59) 10n + 3 n n 2 n=1

∞ 60) 2n 2 4n - 1 n=1 9.6--Review of Tests for Convergence (day #3)

Determine the convergence or divergence of the series. Identify the test used: ∞ 61) cos n 3n n=1

∞ 62) (-1)n

n ln n n=2

∞ 63) n n 7

n ! n=1

∞ 64) ln n n2 n=1