Education.com
Try
Brainzy
Try
Plus

# Convergence Tests for AP Calculus

(not rated)
By McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: Series Practice Problems for AP Calculus

### Integral Test

If an = f (n) where f is a continuous, positive, decreasing function on [c,∞), then the series is convergent if and only if the improper integral f(x) dx exists.

#### Example 1

Determine whether the series converges or diverges.

Step 1:  is continuous, positive, and decreasing on the interval [2,∞).

Step 2:

#### Example 2

Determine whether the series converges or diverges.

Step 1:  is continuous, positive, and decreasing on the interval [1,∞).

Step 2:

Since the improper integral exists, the series converges.

#### Example 3

Determine whether the series converges or diverges.

Step 1:  f (x ) = is continuous, positive, and decreasing on [1,∞).

Step 2:

#### Example 4

Determine whether the series converges or diverges.

Step 1:  f (x ) = is continuous, positive, and decreasing on [1,∞).

Step 2:

### Ratio Test

If , is a series with positive terms and , then the series converges. If the limit is greater than 1 or is ∞, the series diverges. If the limit is 1, another test must be used.

#### Example 1

Determine whether the series converges or diverges.

Step 1: For all is positive.

Step 2:

Since this limit is less than 1, the series converges.

#### Example 2

Determine whether the series converges or diverges.

Step 1: For all is positive.

Step 2:

Since this limit is greater than one, the series diverges.

### Comparison Test

Suppose and are series with non-negative terms, and is known to converge. If a term-by-term comparison shows that for all n, an ≤ b, then converges. If diverges, and if for all n, an, then diverges. Common series that may be used for comparison include the geometric series, which converges for r < 1 and diverges for r ≤ 1, and the p-series, which converges for p > 1 and diverges for p ≤ 1.

#### Example 1

Determine whether the series converges or diverges.

Step 1: Choose a series for comparison. The series can be compared to the p-series with p=2.Both series have non-negative terms.

Step 2: A term-by-term comparison shows that for all values of n.

Step 3: converges, so

#### Example 2

Determine whether the series + … converges or diverges.

Step 1: The series + … = can be compared to .

Step 2: for all n ≤ 1.

Step 3: Since diverges, … also diverges

### Limit Comparison Test

If an and bn are series with positive terms, and if where 0 < L < ∞, then either both series converge or both diverge. By choosing, for one of these, a series that is known to converge, or known to diverge, you can determine whether the other series converges or diverges. Choose a series of a similar form so that the limit expression can be simplified.

#### Example 1

Determine whether the series converges or diverges. The given series . Choose for comparison, since it has a similar structure, and we know it is a p-series with p = 1, it diverges. The limit . The limit exists and is greater than zero; therefore, since = also diverges.

#### Example 2

Determine whether the series converges or diverges.

Compare to the known convergent p-series . The limit . Since 0 < ∞ and converges. also converges.

Practice problems for these concepts can be found at: Series Practice Problems for AP Calculus

150 Characters allowed

### Related Questions

#### Q:

See More Questions

### Today on Education.com

Top Worksheet Slideshows