Find the limit of the sequence given by recurrence relation

Question

Find the limit of the sequence given by recurrence relation $c_{n+1} = (1-\frac{1}{n})c_n + \beta_n $ where $ \beta_n $ is any sequence with the property $n^2\beta_n \to 0$ as $ n \to \infty$

I've proved that $c_{n+1} = \frac{1}{n}\sum\limits_{i=1}^n i\beta_{i}$ but then i got stuck.

Answer 1

I get:
$c_2 = \beta_1$
$c_3 = (1/2)c_2 + \beta_2 = (1/2)\beta_1 + (2/2)\beta_2$
$c_4 = (2/3)c_3 + \beta_3 = (1/3)\beta_1 + (2/3)\beta_2 + (3/3)\beta_3$
$c_5 = (3/4)c_4 + \beta_4 = (1/4)\beta_1 + (2/4)\beta_2 + (3/4)\beta_3 + (4/4)\beta_4$

Which seems to be a different solution than yours.

Also, this will give a similar limit for $c_n$ under the weaker condition $n\beta_n\rightarrow 0$.

---

Hour later: Yes, I see you have edited your question to match the solution suggested above. So this is an average of $i\beta_i$ values.

Answer 2

We have $n c_{n+1} = (n-1) c_n + \gamma_n$ where $\gamma_n = n \beta_n$, and so $n \gamma_n \to 0$

Let $c_n = \frac{1}{n-1}\sum_{j=1}^{n-1} a_j $. Then

$$\sum_{j=1}^{n} a_j = \sum_{j=1}^{n-1} a_j +\gamma_n \implies a_n=\gamma_n$$

Hence $c_n $ is the Cesaro mean of $\gamma_n$. Because $\gamma_n \to 0$, so does $c_n$ ( unless I made some mistake, it seems that the weaker condition $n\beta_n \to 0$ was enough).