That's amazing! I was so focused on thinking of a test that would work that I didn't even pay attention to the very basics! Thanks for giving me this perspective.
Okay...
Now if I want to prove absolute convergence, I want to prove that the series ##\frac 1 {2^2}+\frac 1 {3^2}+\frac 1 { 2^3}+\frac 1 { 3^3}+...## is convergent.
This can be viewed as the sum of two convergent geometric series and so it will be convergent as well proving that my series...
If I group the odd and even terms,I obtain ##\sum_{n=2}^\infty {(1/2^n)-(1/3^n)}##
This is less than the geometric series ##\sum_{n=2}^\infty {(1/2^n)}## which is convergent. This makes my series convergent as well.
However, if the original series were conditionally convergent, wouldn't...
Here is a plot of the magnitudes of a few terms. Although what you said is true( every even term is smaller than the preceeding odd term), every odd term after an even term is larger. How can I tell now whether the series is convergent or divergent?
These are my steps:
I considered the even term as an=-1/(3^(0.5n+1)) which makes the odd term after it an+1=1/(2^(0.5n+2).
The ratio rn=0.75*((1.5)^n) approaches infinity as n approaches infinity. This means that the series is divergent in this case.
I plugged in n+1 for n to determine the odd...
Homework Statement
Test the series for convergence or divergence
##1/2^2-1/3^2+1/2^3-1/3^3+1/2^4-1/3^4+...##
Homework Equations
rn=abs(an+1/an)
The Attempt at a Solution
With some effort I was able to figure out the 'n' th tern of the series
an =
\begin{cases}
2^{-(0.5n+1.5)} & \text{if } n...
Yes. I realize that my solution is wrong. This is not what I intended when I said I understand this. I meant to conserve momentum first and then energy. This is to show what I did wrong to anyone who is curious. Sorry for the confusion.