Originally Posted by

**Henryt999** Are these answers I came up with acctually solutions or more just idéas that doesn´t acctually show anything and by "coincidence" happend to have the same numbers as the actual solutions?

Question is:

Look at $\displaystyle [a_n]_{n=1}^{\infty}$

It is defined by: $\displaystyle a_{n+1} = a_n-a_n^2$ with $\displaystyle a_0$ = $\displaystyle \frac{1}{2}$ and $\displaystyle 0\leq n$

a) Find the limit of the recursive formula.

b)Find the limit of $\displaystyle \lim_{n\rightarrow \infty} \sum_{k=0}^{n}a_k^2$ this one I´m not sure I understand everywhere is $\displaystyle a_n$ suddenly we have $\displaystyle a_k$ and then from k=0 where do I put the 0. Or is it just that the sum from..$\displaystyle a_1$..oh well lets just assume it means from $\displaystyle a_1$

a: Solution: Since the limit of $\displaystyle a_{n+1}$ =limit $\displaystyle a_n$

we can write (I think) $\displaystyle a=a-a^2$

this is only satisfied if a = 0. Hence the limit must be 0....hmm book said 0 too, well then something must be right (maybe)

b) Solution: $\displaystyle \lim_{n\rightarrow \infty} \sum_{k=0}^n a_k^2 = \lim_{n\rightarrow\infty}(\frac{1}{2}-(\frac{1}{n})^n) = \lim_{n\rightarrow \infty} (\frac{1}{2}-\frac{1^n}{n^n}) = \frac{1}{2}$ . Book say 1/2..Wow I must have done something correct, ooor there was a $\displaystyle \frac{1}{2} $ that magicaly appered