# Calculus Examples

Determine if the Series is Divergent
Step 1
The series is divergent if the limit of the sequence as approaches does not exist or is not equal to .
Step 2
Evaluate the limit.
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
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