Series: Converge or Diverge

[KAPPY]

"The terms of a series are defined by the formulas give. Does the series Summation form n=1 to infinity A(sub n) converge, or diverge? Give reasons for your answers."


a(sub 1) = 3 , a(sub n+1) = [n/(n+1)](a(sub n))

I was told there were two ways to solve this problem, either by plugging in numbers and looking for a geometric pattern or by using a convergence/divergence test on the a(sub n+1) formula (usually the ratio test because by definition we divide by a(sub n) so we can just take the limit of what they give us with the a(sub n) divided out).
But the back of the book says we use the integral test on this problem, so how would that work with the a(sub n)?

Am I right in the steps to solve the problem as stated above, or am I missing something? <- for use on other problems

Also, when I tried the ratio test just for the fun of it and I used l'hopitals rule and got 1, so I know that the Ratio Test fails.

 

[DrPhil]

"The terms of a series are defined by the formulas give. Does the series Summation form n=1 to infinity A(sub n) converge, or diverge? Give reasons for your answers."


a(sub 1) = 3 , a(sub n+1) = [n/(n+1)](a(sub n))

I was told there were two ways to solve this problem, either by plugging in numbers and looking for a geometric pattern or by using a convergence/divergence test on the a(sub n+1) formula (usually the ratio test because by definition we divide by a(sub n) so we can just take the limit of what they give us with the a(sub n) divided out).
But the back of the book says we use the integral test on this problem, so how would that work with the a(sub n)?

Am I right in the steps to solve the problem as stated above, or am I missing something? <- for use on other problems

Also, when I tried the ratio test just for the fun of it and I used l'hopitals rule and got 1, so I know that the Ratio Test fails.

In my opinion it is "always" good to write out several terms.
3, 3/2, 3/3, 3/4, 3/5, . . . 3/n
This is a harmonic series. The ratio test = 1 is not sufficient to prove that it diverges - but you can show that it does diverge by grouping terms [that proof may be in your text or notes?].

For the integral test for the partial sums, consider

\(\displaystyle \displaystyle \sum_1^n \dfrac{3}{i} \; \Rightarrow \; \int_1^n \dfrac{3}{x} dx \)

How does that integral behave?

 

[KAPPY]

Thank you for the help