Question

Alternating series test help

Answer 1

Recall that the Alternating Series Test has three conditions associated with it:
1. The series must alternate.
2. The terms must decrease (in absolute value) for large n.
3. The nth term must go to 0.

Can you even have a series where the first two conditions hold and the third doesn't?
Create a convergent series that satisfies conditions 1 and 3, but not 2.
Create a divergent series that satisfies conditions 1 and 3, but not 2

Answer 2

Yep, we just had an example of it in your previous question, I believe.
As for the other two, the first one I don't believe is necessarily possible, but the latter is a complicated series, which can look something like this:
\[
\sum_ip(i)
\]Where:
\[
p(i)
\begin{cases}
0, \text{ if $p$ is not prime}\\
1, \text{ otherwise}
\end{cases}
\]

Answer 3

It works since the terms never decrease, but as the series goes to infinity, the scarcity of primes becomes approximately \(\frac{\ln(x)}{x}\), or, in other words, zero.

Answer 4

where is the nth term.....i ?

Answer 5

I don't quite get what you're asking.

Answer 6

the third possibility says, the nth term must go to zero

Answer 7

It does, read my case, above.

Answer 8

ohk and is it convergent or divergent series

Answer 9

There's still an infinite number of primes, so, divergent.

Answer 10

oh right, and do you have example of convergent

Answer 11

Nope, I don't think it's possible, but I'd love to see one, if it is.

Answer 12

i think somone answered it...lemme give u the link

Answer 13

Awesome.

Answer 14

\[a_n = \frac{ 1 }{ n^2 } \] n is even
\[a_n = - \frac{ 1 }{ n^3 }\] n is odd

Answer 15

But both of those decrease, don't know how that doesn't apply to condition (2).

Answer 16

OH, never mind, that's ONE series.

Answer 17

Yep, that works. Nice. Didn't think of composing two functions.

Answer 18

oh ya i didn't know how to do it here so haha