# Need help with these homework problems

• Aug 4th 2010, 07:45 AM
nxd10
Need help with these homework problems
Hi All,

I took my time to scan these 7 problems from the book to see if you guys can help me out with any of them. I've missed about a week of class due to work travel, and it would really help if you guys can help me with the solutions, as I have a test in 4 days and need to know how to do these.

http://modernvirtual.com/1.png
http://modernvirtual.com/2.png

• Aug 4th 2010, 08:58 AM
yeKciM
let's go from the beginning :D

1. problem

what the criteria for testing convergence do you know? (if u know any try applying it and if u have problem say where )
• Aug 4th 2010, 09:59 AM
nxd10
I honestly don't know even where to start with these. I've missed about five days of a 3 hour class, so that was a lot of material. I tried to look up the material online and tried wolfram alpha as well as trying a few methods by hand and still cannot even comprehend how to do these. :(
• Aug 4th 2010, 10:17 AM
yeKciM
u have D'alamberth (sorry if i mistype his name)

$\displaystyle \sum_{n=1}^{\infty}a_n$

then :

$\displaystyle q=\lim_{n\to \infty} \frac {a_{n+1}}{a_n}$

so u have a cases :

$q=\begin{cases}
q<1 & converges \\
q>1& diverges \\
q=1& ?
\end{cases}$

or Rabel's (again sorry for mistyping) :D

$\displaystyle \sum_{n=1}^{\infty}a_n$

$\displaystyle q=\lim_{n\to \infty} n[\frac {a_{n+1}}{a_n}-1]$

$q=\begin{cases}
q<1 & converges \\
q>1& diverges \\
q=1& ?
\end{cases}$

or u can do it "by definition" :D
definition say that series converges or diverges if

$\displaystyle \lim_{N\to \infty} S_N$

is number then converges
if infinity nuber or doesn't exist it diverges :D

$(S_N)_{N\in\mathbb{N}}$ is array of partial sums
• Aug 4th 2010, 10:56 AM
nxd10
Ok, that helped a ton. I think i have them all figured out except for #3 and #5. How would I go about those?
• Aug 4th 2010, 11:10 AM
yeKciM
Quote:

Originally Posted by nxd10
Ok, that helped a ton. I think i have them all figured out except for #3 and #5. How would I go about those?

first of all those criteria that I wrote there isn't only them.... there are few more :D

as 3. problem i'm trying to google translate it :D (sorry if i understand what's point i'll post it :D )

for 5. problem i think is something to do with "Štolc" theorem (i think is something like "Shtolch" or :D)

he says that :

"if $(x_n)_{n\in\mathbb{N}}$ and $(y_n)_{n\in\mathbb{N}}$ arbitrary real sequences such that a sequence $(y_n)$ satisfies:

$(y_n)$ monotonically increasing

$\displaystyle \lim_{n\to\infty} y_n = \infty$

if exist :

$\displaystyle \lim_{n\to\infty} \frac {x_{n+1}-x_n}{y_{n+1}-y_n}$

then exist and

$\displaystyle \lim_{n\to\infty} \frac {x_n}{y_n}$

and they are equal :D

i think that's it :D

hehehe :D or u can (i think again ) for #5 just (to se if is increasing or decreasing) just put $a_n < a_{n+1}$ and if it's true then is increasing but if $a_n > a_{n+1}$ then is decreasing :D (we do it like that here :D )