The Integral Test:

(i) If is convergent, then is convergent.

(ii) If is divergent, then is divergent.

if the the result is finite it’s convergent

if the the result is infinite it’s divergent

NOTE : When we use the Integral Test, it is not necessary to start the series or the integral at n=1.

Example 1

Test the series for convergence or divergence ?

solution :

The function is continuous, positive, and decreasing on [1,∞ ) so we use the integral test :

Example 2

Test the series for convergence or divergence ?

solution :

The function is continuous, positive, and decreasing on [1,∞ ) so we use the integral test :

Example 3 :

Test the series for convergence or divergence ?

solution :

The function is continuous, positive, and decreasing on [1,∞ ) so we use the integral test :

NOTE :

From integral test the series is convergent .

Example 4 :

Determine whether the series is convergent or divergent ?

solution :

The function is continuous, positive, and decreasing on [1,∞ ) so we use the integral test :

The P-series :

Example 5:

Test the series is converges or diverges?

solution :

It’s convergent because it’s P-series with P=2 >1

Example 6:

Test the series is converges or diverges?

solution :

It’s convergent because it’s P-series with P=3 >1

Series: The Ratio Test

The Integral Test and P-series

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