# Calculus Examples

Step 1

For an infinite series , find the limit to determine convergence using Cauchy's Root Test.

Step 2

Substitute for .

Step 3

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

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

Simplify the answer.

Step 4.7.1

Simplify the numerator.

Step 4.7.1.1

Multiply by .

Step 4.7.1.2

Add and .

Step 4.7.2

Add and .

Step 4.7.3

is approximately which is positive so remove the absolute value

Step 4.8

Divide by .

Step 5

If , the series is absolutely convergent. If , the series is divergent. If , the test is inconclusive. In this case, .

The series is convergent on