1. ## Is this correct?

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

2. Remember the little letters are just index character and do not have a special meaning beyond that. You can use whatever you want.

if k=1,2,... and n=1,2,... then $\displaystyle (a_k)_k=(a_n)_n$

With that said yhou are asked to compute

$\displaystyle \lim_{n\rightarrow \infty} \sum_{k=0}^{n}a_k^2$ dont sweat about it being changed

So yo have that $\displaystyle \lim_{n\rightarrow \infty} \sum_{k=0}^{n}a_k^2 = \lim_{k\rightarrow \infty} \sum_{n=0}^{k}a_n^2$

I would suggest that you use the recursion formula you know to get a telescopic series and see what happens

3. Oh and about a) i think you did right.

4. 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
For the first question, you can only use arithmetic of limits if you proved that the sequence converges, which you did not.

To prove the sequence converges, show that it is bounded and monotonically decreasing. Only after you show that you can proceed to the final step which you described.