`sum_(n=1)^oo (n+2)/(n+1)` Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series.

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gsarora17 eNotes educator| Certified Educator


Integral test is applicable if f is positive, continuous and decreasing function on infinite interval `[k,oo)` where `k>=1` and `a_n=f(x)` . Then the series `sum_(n=1)^ooa_n`  converges or diverges if and only if the improper integral `int_1^oof(x)dx` converges or diverges.

For the given series `a_n=(n+2)/(n+1)`

Consider `f(x)=(x+2)/(x+1)`

Refer to the attached graph of the function. From the graph, we observe that the function is positive , continuous and decreasing for `x>=1`

We can also determine whether the function is decreasing by finding its derivative f'(x).

Apply quotient rule to find f'(x),





`f'(x)<0` which implies that f(x) is decreasing for `x>=1`

We can apply integral test, as the function satisfies all the conditions for the integral test.

Now let's determine whether the corresponding improper integral `int_1^oo(x+2)/(x+1)dx` converges or diverges,


Let's first evaluate the indefinite integral,



Apply the sum rule,








Since the integral `int_1^oo(x+2)/(x+1)dx` diverges, we conclude from the integral test that the series diverges.

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