How can I do this troglodyte of a series

[frasifrasi]

The series, which is by all accounts nefarious, is (-1)^(n)*n*e^(-2^n).

In an attempt to subdue this monster, I made it into (-1)^(n)*n*/e^(2^n), but I do not want to wing the rest of the solution.

So, I ask for your help. Help me, brothers, like there is no tomorrow.

 

[quasar987]

Seems like a good candidate for Leibniz criterion of alternate series. If the general term in absolute value decrease and goes to zero, then the series converges.

 

[DaveC426913]

BTW, troglodyte simply means underground dweller.

 

[frasifrasi]

Thanks, but I need to know why it converges absolutely, which is the case...

 

[quasar987]

Can't you think of a convergence test you could apply on n*e^(-2^n) ?

 

[frasifrasi]

if tha absolute value converges...

n/e^(2n), I would assume e grows faster since it is an exponential function...so the abs value converges?

 

[quasar987]

That's an intuitive argument, not a mathematical proof.

(And the argument is flawed too. For instance, for the series of 1/n. We have that 1/n-->0, but the series of 1/n still diverges.)

 

[frasifrasi]

Ok, so how else can I show that it is abs convergent? How should I demonstrate this?

 

[quasar987]

There is a plethora of citerion to determine the convergence of a series. Have you tried them all?

 

[Saladsamurai]

The vocabulary in this thread ROCKS!

Casey