Question

series questions convergence divergence

Answer 1

more of them?

Answer 2

yes they never end

Answer 3

did you run the absolute test?

Answer 4

if sum |an| is convergent, then it is absolutely convergent; if sun is convergent but |an| diverges, then it is conditional

Answer 5

\[4.\\
\sum_{n=1}^\infty (-1)^n\frac{1}{\sqrt n}\]
does not converge absolutely because \(\displaystyle\sum_{n=1}^\infty\left|\frac{(-1)^n}{\sqrt n}\right|\) is not finite (\(p\)-series with \(p=1/2\)).

Answer 6

so it converges conditionally then?

Answer 7

Do you know whether \(\sum a_n\) converges? If so, then yes, the series is conditionally convergent. Otherwise it diverges.

Answer 8

For \(\sum a_n\) to converge, by the alternating series test, you must have that
\[\lim_{n\to\infty}\frac{n^n}{n!}=0\]
and that \(\dfrac{n^n}{n!}\) is a decreasing sequence. Do these two hold?

Answer 9

Oh sorry, I'm looking at the wrong series...

Answer 10

You still have to satisfy the above conditions for the FIRST series to be conditionally convergent.
\[\lim_{n\to\infty}\frac{1}{\sqrt n}=0~~\checkmark\]
\[\text{Decreasing?}~~\frac{1}{\sqrt1}>\frac{1}{\sqrt2}>\cdots~~\checkmark\]
So yes, this series is conditionally convergent.

Answer 11

For the second series, you're right about the divergence, but apparently you can obtain this result from all the listed tests (other than comparison, I think, unless you can think of a proper comparison series).