Determine which of the following sequences, , converge and find the limit of those that do
i)
ii)
If $\displaystyle \lim_{n\to\infty} a_n$ does not exist or if $\displaystyle \lim_{n\to\infty} a_n \not{=} 0$, then the series $\displaystyle \sum_{n=1}^\infty a_n$ is divergent.
(this should be rather obvious, if you are continuing to add a number infinity times, and it is not zero, it will diverge towards infinity)
ok this is fairly simple
i)$\displaystyle a_n=\frac{n^2}{15n+5}$...so we need to take the limit to see if it diverges by the n-th term test...o wait we just want diverging sequences so we will take the limit and see if we get a non-∞ answer...so we have $\displaystyle \lim_{n \to {\infty}}\frac{n^2}{15n+5}$..multiplying top and bottom by $\displaystyle \frac{\frac{1}{n}}{\frac{1}{n}}$ or using L'hopitals rule the answer is obviously ∞ which means the series is divergent...
ii) we have that $\displaystyle \lim_{n \to {\infty}}(n^2+n)^{\frac{1}{n}}$...now if we say the limit is equal to y we have $\displaystyle y=\lim_{n \to {\infty}}(n^2+n)^{\frac{1}{n}}$...so then using the everuseful rules of logarithims we have that $\displaystyle ln(y)=\lim_{n \to {\infty}}\frac{1}{n}\cdot\ln(n^2+n)=\lim_{n \to {\infty}}\frac{1}{n}\cdot\ln(n(n+1))=\lim_{n \to {\infty}}\frac{1}{n}\cdot[ln(n)+ln(n+1)]$$\displaystyle =\lim_{n \to {\infty}}\frac{ln(n)}{n}+\frac{ln(n+1)}{n}$...now using lhopitals rule we get $\displaystyle ln(y)=\lim_{n \to {\infty}}\frac{\frac{1}{n}}{1}+\lim_{n \to {\infty}}\frac{\frac{1}{n+1}}{1}=\lim_{n \to {\infty}}\frac{1}{n}+\lim_{n \to {\infty}}\frac{1}{n+1}=0$...so we have $\displaystyle ln(y)=0$ and since y was the original limit we want it by itself..so we have $\displaystyle y=e^0=1$...so the limit converges to the value of e...just so you know I showed you the long way instead of seperating you could have just used l'hopitals rule when you still had $\displaystyle ln(n^2+n)$...I thought I would just take the long road
Stop using l'Hôpital's rule !! >_<
It annihilates other methods of calculating limits !
$\displaystyle \lim_{n \to {\infty}}\frac{1}{n}\cdot[ln(n)+ln(n+1)]$
This can be solved by comparing the increasings...
Wikipedia i'm sorry, i can't find any equivalent in english as i don't know the name... It's in "énoncé des résultats..."
Sure, l'Hôpital's rule does it all. But also stop thinking that because it's simple for you that it'll be simple for everybody.ok this is fairly simple
By the way, i think we're rather talking about series than sequences, as it's $\displaystyle \left\{a_n\right\}$ we're studying.
I know it can but for the general kid that just wants help understanding the basics of limits...then comparison of increasing would just confuse them generally...I am trying to teach them what they need to pontificate upon advanced mathematical techniques they may or may not understand. Is that ok with you?