Infinity Expressions

[tonycashflow]

I am having a hard time determining the result of expression when there a infinity.

For example this expression :

I cannot find what this equals to...

The only thing I know is that 1/infinity = 0
but I don't know what, for example, x^infinity equals to, nor ln(infinity) or e^infinity.

Could some help me out with that ?

 

[tkhunny]

:shock: :shock: :shock: The expression you have written is pitifully meaningless. Never, ever, write anything like that again.

Can you provide the actual problem statement?

 

[tonycashflow]

Ok lol. Sorry about that. I'll explain how I came up with that.

I had this :

and I needed to find the interval of convergence.

I used this formula : lim n-->infinity Absolute value of (An+1)/An , to find the radius of convergence.
Doing the division and replacing n by infinity I cam up with the expression in my previous post.

What did I do wrong ?

 

[galactus]

Did you try the ratio test for convergence?.

\(\displaystyle \frac{(n+1)x^{n+1}}{(n+1)^{3}+1}\cdot\frac{n^{3}+1}{nx^{n}}=\frac{(n+1)(n^{3}+1)x}{n^{4}+3n^{3}+3n^{2}+2n}\)

If we take the limit of \(\displaystyle {\rho}=\lim_{n\to{\infty}}|\frac{u_{k+1}}{u_{k}}|\), we get |x|.

This implies that the series converges if \(\displaystyle {\rho}=|x|<1\) and diverges if \(\displaystyle {\rho}=|x|>1\)

The test is inconclusive if |x|=1 (i.e., if x=1 or x=-1)

So, convergence at these points must be investigated separately.

 

[tonycashflow]

Can you explain the part where you take the limit to the expression, which gives you |x|, because this is the part I don't understand...I understand why you do it, but I don't understand how to do it.
Also, what would be the interval of convergence ?

thanks

 

[galactus]

When you take the limit as n --> infinity of a rational function, you can disregard all but the highest terms.

You get \(\displaystyle \frac{n^{4}}{n^{4}}=1\). Therefore, you end up with x.