# Calculus Examples

Determine if Convergent Using Cauchy's Root Test
Step 1
For an infinite series , find the limit to determine convergence using Cauchy's Root Test.
Step 2
Substitute for .
Step 3
Simplify.
Step 3.1
Move the exponent into the absolute value.
Step 3.2
Multiply the exponents in .
Step 3.2.1
Apply the power rule and multiply exponents, .
Step 3.2.2
Cancel the common factor of .
Step 3.2.2.1
Cancel the common factor.
Step 3.2.2.2
Rewrite the expression.
Step 3.3
Simplify.
Step 4
Evaluate the limit.
Step 4.1
Move the limit inside the absolute value signs.
Step 4.2
Divide the numerator and denominator by the highest power of in the denominator, which is .
Step 4.3
Evaluate the limit.
Step 4.3.1
Simplify each term.
Step 4.3.1.1
Cancel the common factor of and .
Step 4.3.1.1.1
Factor out of .
Step 4.3.1.1.2
Cancel the common factors.
Step 4.3.1.1.2.1
Factor out of .
Step 4.3.1.1.2.2
Cancel the common factor.
Step 4.3.1.1.2.3
Rewrite the expression.
Step 4.3.1.2
Cancel the common factor of .
Step 4.3.1.2.1
Cancel the common factor.
Step 4.3.1.2.2
Rewrite the expression.
Step 4.3.2
Cancel the common factor of .
Step 4.3.2.1
Cancel the common factor.
Step 4.3.2.2
Divide by .
Step 4.3.3
Split the limit using the Limits Quotient Rule on the limit as approaches .
Step 4.3.4
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 4.3.5
Move the term outside of the limit because it is constant with respect to .
Step 4.4
Since its numerator approaches a real number while its denominator is unbounded, the fraction approaches .
Step 4.5
Evaluate the limit.
Step 4.5.1
Evaluate the limit of which is constant as approaches .
Step 4.5.2
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 4.5.3
Evaluate the limit of which is constant as approaches .
Step 4.6
Since its numerator approaches a real number while its denominator is unbounded, the fraction approaches .
Step 4.7
Step 4.7.1
Simplify the numerator.
Step 4.7.1.1
Multiply by .
Step 4.7.1.2