# Sequences and Series - Power Series Question

• Apr 20th 2008, 07:19 PM
Schniz2
Sequences and Series - Power Series Question
Hey, i have a tonne of these type of questions and i cannot for the life of me understand how they are done....
This is the answer to one of them which asks you to find the convergence and interval of convergence of the series: The sum from n=1 to infinity of the A(n) term at the start of the solution. I don't understand the step where they replace lim{[(n)/(n+1)]^3} with 1^3.

Not sure if i grasp the concept of what to do (Headbang)... can anyone explain?

Thanks.
• Apr 20th 2008, 07:25 PM
Jhevon
Quote:

Originally Posted by Schniz2
Hey, i have a tonne of these type of questions and i cannot for the life of me understand how they are done....
This is the answer to one of them which asks you to find the convergence and interval of convergence of the series: The sum from n=1 to infinity of the A(n) term at the start of the solution. I don't understand the step where they replace lim{[(n)/(n+1)]^3} with 1^3.

Not sure if i grasp the concept of what to do (Headbang)... can anyone explain?

Thanks.

Um, $\lim_{n \to \infty} \frac n{n + 1} = 1$

there are many ways to see this. first you can realize that as $n \to \infty$, + 1 does not matter, so the fraction behaves like n/n = 1

you could also realize that multiplying the numerator and denominator by 1/n yields $\frac 1{1 + \frac 1n}$ which goes to 1 as $n \to \infty$, since $\lim_{n \to \infty} \frac 1n = 0$

you can also use L'Hopital's rule. there are many ways to see this
• Apr 20th 2008, 07:30 PM
Schniz2
Quote:

Originally Posted by Jhevon
Um, $\lim_{n \to \infty} \frac n{n + 1} = 1$

there are many ways to see this. first you can realize that as $n \to \infty$, + 1 does not matter, so the fraction behaves like n/n = 1

you could also realize that multiplying the numerator and denominator by 1/n yields $\frac 1{1 + \frac 1n}$ which goes to 1 as $n \to \infty$, since $\lim_{n \to \infty} \frac 1n = 0$

you can also use L'Hopital's rule. there are many ways to see this

gosh, sometimes i wonder if i still have a brain. i've always suckes at limits and sequences.

Thanks for that.
• Apr 20th 2008, 07:32 PM
Jhevon
Quote:

Originally Posted by Schniz2
gosh, sometimes i wonder if i still have a brain. i've always suckes at limits and sequences.

Thanks for that.

it is ok. i wonder the same thing about myself at times...