Calculus Examples

Determine Convergence with the Integral Test
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
Simplify the answer.
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Step 5.1
Evaluate at and at .
Step 5.2
Remove parentheses.
Step 6
Evaluate the limit.
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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.
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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 .
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Multiply by .
Multiply by .
Step 6.4.4
Combine the numerators over the common denominator.
Step 6.4.5
Simplify the numerator.
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Move to the left of .
Subtract from .
Step 7
Since the integral is convergent, the series is convergent.
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