# Thread: few simple limit problems :D

1. ## 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...

(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

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 ... just with numbers like

\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" hope it's correct

thank you all very very much

2. In the first question I like your idea of evaluating it as two separate series, but they are arithmetic series.

For the odd terms 1 + 3 + 5 + ... there would be n/2 terms, and for -(2 + 4 + 6 + ...) there would be n/2 terms. Use S = N[2a + (N - 1)d]/2 for each.

3. Thank you

perhaps it's stupid question but I'm not familiar with what do represent there "2a" and "d" what you have wrote there

Edit :

I have found in one of the books that Stolz th goes like this ...

\lim_{n\to \infty} \frac{x_n - x_{n-1}}{y_n - y_{n-1}} ... and like that i get second ok does it matter which do I use ? or are they the same

4. a is the first term and d is the common difference.

5. hmmm I think I may have done another stupid mistake

for (1+3+5+..... ) S_(odd)= N[2*1 + (N-1)*2] /2 = .... = N^2
for -(2+4+6+.... ) S_(even) = N[2*2 + (N-1)*2]/2 = .... = N^2 +N

so combined S_(odd) - S_(even) = -N

\lim_{n\to \infty} \frac{-n}{n}

and than I get limit to be -1 ?

6. No, N and n are not the same thing. I've used N to represent the number of terms in the smaller series, while n represents the number of terms in the alternating series.

So N = n/2.

7. aaah I got it thank you very very much