# Will this series converge? If so, what is its limit?

If $a_n=(a_{n-1}+a_{n-2})/2$ and $a_1, a_2$ are given, will this series converge? And if so, what is the limit?

By intuition I think it converges to $(a_1+2a_2)/3$ , but I am not able to prove it.

• Try to determine the $n$th term in the sequence in terms of $n$, $a_1$ and $a_2$. You should notice that $a_n=f(n)a_1+g(n)a_2$, where $f,g$ are some functions related to $2^n$ (and you can prove this by induction). Try to prove that $f(n)\to\frac{1}{3}$ and $g(n)\to\frac{2}{3}$. – Peter Huxford Sep 12 '14 at 7:35
• Please use the word sequence instead of series. They have different meanings in maths. – user37238 Sep 12 '14 at 8:17

## 1 Answer

We have $$2a_n-a_{n-1}-a_{n-2}=0$$

Using Recurrence Relation formula , $$a_n=A+B\left(-\frac12\right)^n$$