# Calculus Examples

Determine Convergence with the Integral Test
Step 1
Check if the function is continuous over the summation bounds.
Step 1.1
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Interval Notation:
Set-Builder Notation:
Step 1.2
is continuous on .
Step 2
Check if the function is positive over the bounds.
Step 2.1
Set up an inequality.
Step 2.2
Solve the inequality.
Step 2.2.1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.2.2
Set equal to .
Step 2.2.3
Set equal to and solve for .
Step 2.2.3.1
Set equal to .
Step 2.2.3.2
Solve for .
Step 2.2.3.2.1
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 2.2.3.2.2
The equation cannot be solved because is undefined.
Undefined
Step 2.2.3.2.3
There is no solution for
No solution
No solution
No solution
Step 2.2.4
The final solution is all the values that make true.
Step 2.2.5
The solution consists of all of the true intervals.
Step 3
Determine where the function is decreasing.
Step 3.1
Write as a function.
Step 3.2
Find the first derivative.
Step 3.2.1
Find the first derivative.
Step 3.2.1.1
Differentiate using the Product Rule which states that is where and .
Step 3.2.1.2
Differentiate using the chain rule, which states that is where and .
Step 3.2.1.2.1
To apply the Chain Rule, set as .
Step 3.2.1.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 3.2.1.2.3
Replace all occurrences of with .
Step 3.2.1.3
Differentiate using the Power Rule which states that is where .
Step 3.2.1.4
Raise to the power of .
Step 3.2.1.5
Raise to the power of .
Step 3.2.1.6
Use the power rule to combine exponents.
Step 3.2.1.7
Simplify the expression.
Step 3.2.1.7.1
Step 3.2.1.7.2
Move to the left of .
Step 3.2.1.8
Differentiate using the Power Rule which states that is where .
Step 3.2.1.9
Multiply by .
Step 3.2.1.10
Simplify.
Step 3.2.1.10.1
Reorder terms.
Step 3.2.1.10.2
Reorder factors in .
Step 3.2.2
The first derivative of with respect to is .
Step 3.3
Set the first derivative equal to then solve the equation .
Step 3.3.1
Set the first derivative equal to .
Step 3.3.2
Factor out of .
Step 3.3.2.1
Factor out of .
Step 3.3.2.2
Multiply by .
Step 3.3.2.3
Factor out of .
Step 3.3.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3.3.4
Set equal to and solve for .
Step 3.3.4.1
Set equal to .
Step 3.3.4.2
Solve for .
Step 3.3.4.2.1
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 3.3.4.2.2
The equation cannot be solved because is undefined.
Undefined
Step 3.3.4.2.3
There is no solution for
No solution
No solution
No solution
Step 3.3.5
Set equal to and solve for .
Step 3.3.5.1
Set equal to .
Step 3.3.5.2
Solve for .
Step 3.3.5.2.1
Subtract from both sides of the equation.
Step 3.3.5.2.2
Divide each term in by and simplify.
Step 3.3.5.2.2.1
Divide each term in by .
Step 3.3.5.2.2.2
Simplify the left side.
Step 3.3.5.2.2.2.1
Cancel the common factor of .
Step 3.3.5.2.2.2.1.1
Cancel the common factor.
Step 3.3.5.2.2.2.1.2
Divide by .
Step 3.3.5.2.2.3
Simplify the right side.
Step 3.3.5.2.2.3.1
Move the negative in front of the fraction.
Step 3.3.5.2.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3.3.5.2.4
Simplify .
Step 3.3.5.2.4.1
Rewrite as .
Step 3.3.5.2.4.1.1
Rewrite as .
Step 3.3.5.2.4.1.2
Rewrite as .
Step 3.3.5.2.4.2
Pull terms out from under the radical.
Step 3.3.5.2.4.3
One to any power is one.
Step 3.3.5.2.4.4
Rewrite as .
Step 3.3.5.2.4.5
Any root of is .
Step 3.3.5.2.4.6
Multiply by .
Step 3.3.5.2.4.7
Combine and simplify the denominator.
Step 3.3.5.2.4.7.1
Multiply by .
Step 3.3.5.2.4.7.2
Raise to the power of .
Step 3.3.5.2.4.7.3
Raise to the power of .
Step 3.3.5.2.4.7.4
Use the power rule to combine exponents.
Step 3.3.5.2.4.7.5
Step 3.3.5.2.4.7.6
Rewrite as .
Step 3.3.5.2.4.7.6.1
Use to rewrite as .
Step 3.3.5.2.4.7.6.2
Apply the power rule and multiply exponents, .
Step 3.3.5.2.4.7.6.3
Combine and .
Step 3.3.5.2.4.7.6.4
Cancel the common factor of .
Step 3.3.5.2.4.7.6.4.1
Cancel the common factor.
Step 3.3.5.2.4.7.6.4.2
Rewrite the expression.
Step 3.3.5.2.4.7.6.5
Evaluate the exponent.
Step 3.3.5.2.4.8
Combine and .
Step 3.3.5.2.5
The complete solution is the result of both the positive and negative portions of the solution.
Step 3.3.5.2.5.1
First, use the positive value of the to find the first solution.
Step 3.3.5.2.5.2
Next, use the negative value of the to find the second solution.
Step 3.3.5.2.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 3.3.6
The final solution is all the values that make true.
Step 3.4
There are no values of in the domain of the original problem where the derivative is or undefined.
No critical points found
Step 3.5
No points make the derivative equal to or undefined. The interval to check if is increasing or decreasing is .
Step 3.6
Substitute any number, such as , from the interval in the derivative to check if the result is negative or positive. If the result is negative, the graph is decreasing on the interval . If the result is positive, the graph is increasing on the interval .
Step 3.6.1
Replace the variable with in the expression.
Step 3.6.2
Simplify the result.
Step 3.6.2.1
Simplify each term.
Step 3.6.2.1.1
One to any power is one.
Step 3.6.2.1.2
Multiply by .
Step 3.6.2.1.3
One to any power is one.
Step 3.6.2.1.4
Simplify.
Step 3.6.2.1.5
One to any power is one.
Step 3.6.2.1.6
Simplify.
Step 3.6.2.2