Does the Nth Term Test Indicate Divergence for \(\sum (1-\frac{3}{n})^n\)?

[kuahji]

For a problem such as \sum (1-3/n)^n with the sum going from 1 to infinity, the textbook shows to use the nth term test & that it diverges. However in a previous chapter the textbook had a theorem that said the limit of \sum (1+x/n)^n will converge to e^x for any number x.

So I'm confused. Are they using convergence in a different context or what am I missing?

 

[quasar987]

It is (1+x/n)^n that converges to e^x, not the sum of that.

And actually, you can use this fact to show that the series in your problem diverges. You know that (1-3/n)^n converges to e^{-3}, which is not 0, hence the series diverges.

 

[kuahji]

Thanks for the reply. I see that I obviously got confused ^_^.