Question

So of course my professor didnt post an answer sheet for this worksheet so if someone can just check my answers that would be awesome:)

*Does the sum converge or diverge?

Sum of pi^n n=1 and b=infinity

I said it converges and i used the integral test

Answer 1

\[\Large \sum_{1}^{\infty} \pi^n\]

Answer 2

yup:)

Answer 3

It doesn't look like it'd converge... what makes you think it will?

You're adding: \[\LARGE \pi +\pi^{2}+\pi^{3}+\pi^{4}...\]

Answer 4

oh that looks much easier than what i did lol

Answer 5

so why do u say it diverges though?

Answer 6

I'm not sure what kinda test you'd need to prove it. But the terms approach infinity as n approaches infinity.

Answer 7

so if the terms approach infinity and doesnt give me a real number basically it diverges?

Answer 8

Yep, always, because you're adding terms that are getting increasingly larger, as opposed to terms approaching zero (which is the only way it can converge, if the terms never approach zero, then you're endlessly adding numbers).

But note that the terms approaching zero doesn't GUARANTEE convergence, it's just one requirement. For example
\[\Large \sum_{1}^{\infty}\frac{ 1 }{ n }\] terms approach zero as n approaches inf, but the sum diverges.

Answer 9

that makes perfect sense! thank you:)

Answer 10

Welcome :) Basically if the terms approach inf, or some number other than zero, the sum never converges. The terms might converge to some number, but then the sum is still adding that number endlessly.

Terms have to approach zero for convergence to even be possible.