few simple limit problems :D

hello,

I'm having problems with few limits... I have done them but my solution isn't the same as in the book... so can anyone check them or point me what do I do wrong... :D

(sorry but LATEX don't work :( I'll fix it when it does)

**first problem ::**

\lim_{n\to \infty} \frac{1-2+3- ... + (-1)^(n-1) n}{ n}

I try using Stolz th but come to way wron solution...

so I look at the 1-2+3- .... as two series ... one with odd and one with even numbers like this :

\sum_{n=0} ^{\infty} (2n+1) - \sum_{n=0} ^{\infty} 2n

so i find nth partial sum of each of them and got this

(n+1)^2 (from the odd ) and n(n+1) (from the even numbers)

so combined those two i get to nth partial sum be just (n+1 )

after that limit looks like :

\lim_{n\to \infty} \frac {n+1}{n}

and solution is 1 but that's not true :D

**second problem is :**

show that

\lim_{n\to \infty} \sqrt[n] {a_1 * a_2 * .... a_n } = \lim_{n\to \infty} a_n

hmmmm here I say that some A_n is A_n = \sqrt[n] {a_1 * a_2 * .... a_n }

so now I do log

\ln (A_n) = \frac{1}{n} * (\ln(a_1) + \ln(a_2) +.... +\ln(a_n)

so i use Stolz th

\lim_{n\to \infty} \frac {x_(n+1) - x_n}{y_{n+1}-y_n}

and I get to be equal to \lim_{n\to \infty} a_(n+1) not a_n (since all another just substract.. )

**Third problem : **

is the same as second ... :D just with numbers like :D

\lim_{n\to \infty} \sqrt[n] {(1+1/1)^1 * (1+1/2)^2 * .... (1+1/n)^n }

and I get to result be "e" :D hope it's correct ;)

thank you all (Bow)(Bow)(Bow) very very much ;)