Convergence & Sum of Alternating Series | Homework Help

[rcmango]

Homework Statement

Determine wheter the series is convergent or divergent. If it convergent, approximate the sum of the series correct to four decimal places.

heres the equation: http://img251.imageshack.us/img251/2261/46755781zg9.png

Homework Equations

The Attempt at a Solution

This appears to be an alternating geometric series,

Would it be okay to move the exponent k over everything? in other words: ( (-1)/k) )^k

So then it looks a lot like a geometric series, so then It converges by the rules of an alernating series, it is decreasing and it is approaching zero.

So then to find its sum, i would do so by geometric series right?

first term would be starting at k = 2, so: 1/2?

then use 1/2 divided by 1 -r

Am i on the right track? what is r? is it also, 1/2?

 

[Pyrrhus]

Yes you can use k as the exponent of the whole because of the distributive property of exponentiation.

 

[rcmango]

how about the rest of what I'm doing here, this was my best hypothesis to approach the problem. I need help with the common ratio. I'm not sure what to use if its k^k ?

 

[dynamicsolo]

Determine wheter the series is convergent or divergent. If it convergent, approximate the sum of the series correct to four decimal places.

heres the equation: http://img251.imageshack.us/img251/2261/46755781zg9.png

This appears to be an alternating geometric series...

It isn't a geometric series because such series has a constant ratio between successive terms. However, that gives you a clue to the proof of its convergence. (Try a comparison test.) As for the estimate of the sum, do they want an analytical proof of some sort or just something carried out on a calculator (how many terms do you need to get to a precision of 10^-4 ?)