1. ## Inconclusive

$\sum_{n=1}^\infty (-1)^{n-1}\frac{\sqrt{n}}{ n+2 }$

I tried the ratio test and I got 1, which means inconclusive. What can I do to tell whether it is converging or diverging?

2. Originally Posted by fastcarslaugh
$\sum_{n=1}^\infty (-1)^{n-1}\frac{\sqrt{n}}{ n+2 }$

I tried the ratio test and I got 1, which means inconclusive. What can I do to tell whether it is converging or diverging?
nth term goes to zero ... terms of the series alternate in sign ... what does that tell you?

3. Originally Posted by fastcarslaugh
$\sum_{n=1}^\infty (-1)^{n-1}\frac{\sqrt{n}}{ n+2 }$

I tried the ratio test and I got 1, which means inconclusive. What can I do to tell whether it is converging or diverging?
Let $a_n=\frac{\sqrt{n}}{n+2}$. Note that $\forall{n}\in\mathbb{N}~a_{n+1}\leqslant{a_n}$. Further note that $\lim_{n\to\infty}a_n=0$. So this series converges by Leibniz's Criterion. Note that although the Ratio Test is very useful, a lot of kids get in a Ratio Test only mindframe, try not to do that.

4. $\sum_{n=1}^\infty (-1)^n\frac{n}{ 5 + \ln n }$

so this is also alternating. and it looks like it's oscillating. Therefore, it's diverging?

Also
$\sum_{n=1}^\infty (-1)^n\frac{ 4 n }{ 8 n + 7 }$

This one is oscillating. it means diverging, right?

5. Originally Posted by fastcarslaugh
$\sum_{n=1}^\infty (-1)^n\frac{n}{ 5 + \ln n }$

so this is also alternating. and it looks like it's oscillating. Therefore, it's diverging?

Also
$\sum_{n=1}^\infty (-1)^n\frac{ 4 n }{ 8 n + 7 }$

This one is oscillating. it means diverging, right?
In this case oscillating and alternating are synonomous. And secondly no that is completely incorrect. Holistically being alternating greatly increases the chances a series converges. It mus satisfy much less stringent requirements. Do you know the Alternating Series Test (Leibniz's Criterion)?

6. Originally Posted by Mathstud28
In this case oscillating and alternating are synonomous. And secondly no that is completely incorrect. Holistically being alternating greatly increases the chances a series converges. It mus satisfy much less stringent requirements. Do you know the Alternating Series Test (Leibniz's Criterion)?

No, Could you explain it?

$
\sum_{n=1}^\infty (-1)^n\frac{n}{ 5 + \ln n }
$

Although this seems to bounce back and forth in sign, it looks like it's approaching a certain value.

7. Originally Posted by fastcarslaugh
No, Could you explain it?
The Alternating Series Test (Lebniz's Criterion) loosely states: If $\sum{(-1)^na_n}$ is an infinite series it converges iff $\lim_{n\to\infty}a_n=0$ and $a_n\in\downarrow$ or if you prefer $a_{n+1}\leqslant{a_n}$

8. Pauls Online Notes : Calculus II - Alternating Series Test

you're not an "online" student, are you?

9. Originally Posted by fastcarslaugh
$\sum_{n=1}^\infty (-1)^{n-1}\frac{\sqrt{n}}{ n+2 }$

I tried the ratio test and I got 1, which means inconclusive. What can I do to tell whether it is converging or diverging?
Try the alternating series test for conditional convergence.

Never mind way to late I guess