1. ## Sequence

Say I have a sequence where for each positive integer n, $\displaystyle a_{n+1}=a_n- \frac {1}{n}$. Is there an $\displaystyle a_1$ I can choose which will make this a convergent sequence?

It seems to me that if one choice of $\displaystyle a_1$ will work, any choice will work, but that's just my intuition. I'm not sure how to go about proving it.

2. Originally Posted by spoon737
Say I have a sequence where for each positive integer n, $\displaystyle a_{n+1}=a_n- \frac {1}{n}$. Is there an $\displaystyle a_1$ I can choose which will make this a convergent sequence?

It seems to me that if one choice of $\displaystyle a_1$ will work, any choice will work, but that's just my intuition. I'm not sure how to go about proving it.
What have you gotten from this recurrence relationship? Have you used it to find anything yet?

3. Originally Posted by spoon737
Say I have a sequence where for each positive integer n, $\displaystyle a_{n+1}=a_n- \frac {1}{n}$. Is there an $\displaystyle a_1$ I can choose which will make this a convergent sequence?

It seems to me that if one choice of $\displaystyle a_1$ will work, any choice will work, but that's just my intuition. I'm not sure how to go about proving it.
Note that if $\displaystyle a_1=a$ then $\displaystyle a_n = a - \sum_{k=1}^n \frac{1}{k}$. This leads to a harmonic series.

4. Oh wow, I can't believe I missed that. So it can't converge at all then.