# Calculus Examples

Step 1

The series is divergent if the limit of the sequence as approaches does not exist or is not equal to .

Step 2

Step 2.1

Divide the numerator and denominator by the highest power of in the denominator, which is .

Step 2.2

Evaluate the limit.

Step 2.2.1

Simplify each term.

Step 2.2.1.1

Cancel the common factor of and .

Step 2.2.1.1.1

Factor out of .

Step 2.2.1.1.2

Cancel the common factors.

Step 2.2.1.1.2.1

Factor out of .

Step 2.2.1.1.2.2

Cancel the common factor.

Step 2.2.1.1.2.3

Rewrite the expression.

Step 2.2.1.2

Cancel the common factor of .

Step 2.2.1.2.1

Cancel the common factor.

Step 2.2.1.2.2

Rewrite the expression.

Step 2.2.1.3

Multiply by .

Step 2.2.2

Cancel the common factor of .

Step 2.2.2.1

Cancel the common factor.

Step 2.2.2.2

Divide by .

Step 2.2.3

Split the limit using the Limits Quotient Rule on the limit as approaches .

Step 2.2.4

Split the limit using the Sum of Limits Rule on the limit as approaches .

Step 2.2.5

Move the term outside of the limit because it is constant with respect to .

Step 2.3

Since its numerator approaches a real number while its denominator is unbounded, the fraction approaches .

Step 2.4

Evaluate the limit.

Step 2.4.1

Evaluate the limit of which is constant as approaches .

Step 2.4.2

Split the limit using the Sum of Limits Rule on the limit as approaches .

Step 2.4.3

Evaluate the limit of which is constant as approaches .

Step 2.4.4

Move the term outside of the limit because it is constant with respect to .

Step 2.5

Since its numerator approaches a real number while its denominator is unbounded, the fraction approaches .

Step 2.6

Simplify the answer.

Step 2.6.1

Simplify the numerator.

Step 2.6.1.1

Multiply by .

Step 2.6.1.2

Multiply by .

Step 2.6.1.3

Subtract from .

Step 2.6.2

Simplify the denominator.

Step 2.6.2.1

Multiply by .

Step 2.6.2.2

Add and .

Step 2.6.3

Move the negative in front of the fraction.

Step 3

The limit exists and does not equal , so the series is divergent.