# Calculus Examples

Step 1

To determine if the series is convergent, determine if the integral of the sequence is convergent.

Step 2

Write the integral as a limit as approaches .

Step 3

Rewrite as .

Step 4

The integral of with respect to is .

Step 5

Step 5.1

Evaluate at and at .

Step 5.2

Remove parentheses.

Step 6

Step 6.1

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

Step 6.2

The limit as approaches is .

Step 6.3

Evaluate the limit of which is constant as approaches .

Step 6.4

Simplify the answer.

Step 6.4.1

The exact value of is .

Step 6.4.2

To write as a fraction with a common denominator, multiply by .

Step 6.4.3

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Step 6.4.3.1

Multiply by .

Step 6.4.3.2

Multiply by .

Step 6.4.4

Combine the numerators over the common denominator.

Step 6.4.5

Simplify the numerator.

Step 6.4.5.1

Move to the left of .

Step 6.4.5.2

Subtract from .

Step 7

Since the integral is convergent, the series is convergent.