Integral Test

Equation
$$\sum_{n = 0}^{oo}a_{n}$$

Use
The Integral Test can only be used if the following are true.$$f(x)$$ is decreasing on the interval $$[n, oo)$$$$f(x)$$ is positive on the interval $$[n, oo)$$$$f(x)$$ is continuous on the interval $$[n, oo)$$If $$f(x)$$ satisfies all of these conditions, take the indefinite integral of the function.The convergence of the series is the same as that of the indefinite integral.</ol>

Explanation
This proof uses the Direct Comparison Test to compare the term $$f(n)$$ with the integral of $$f(x)$$ over the distance of $$[n,n+1)$$. Since we have already established the three conditions above we can say that $$f(x)$$ converges through the Direct Comparison Test.

Video Explanation


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