1. ## Proof of Convergence

Prove that $\displaystyle \sum_{n=1}^\infty\!a_n$ converges if and only if $\displaystyle \sum_{n=N}^\infty\!a_n$ converges for any $\displaystyle N \geq 1$

My approach..
Assume that the latter series converges then this inequality holds for some real $\displaystyle M$
$\displaystyle \left| \sum_{n=1}^\infty\!a_n - \sum_{n=N}^\infty\!a_n \right| < M$ for any $\displaystyle N \geq 1$

And thus does also the first series converge. They only differ by a constant depending on $\displaystyle N$.

Now assume that $\displaystyle \sum_{n=N}^\infty\!a_n$ diverges to $\displaystyle \pm\infty$ for some $\displaystyle N > 1$ and that $\displaystyle \sum_{n=1}^\infty\!a_n$ converges.

$\displaystyle \sum_{n=1}^\infty\!a_n = \sum_{n=1}^{N-1}a_n + \sum_{n=N}^\infty\!a_n$

This inequality holds for some $\displaystyle M$
$\displaystyle \left| \sum_{n=1}^\infty\!a_n - \sum_{n=1}^{N-1}\!a_n \right| < M$ for any $\displaystyle N \geq 1$
But this inequality doesn't hold for any $\displaystyle M$
$\displaystyle \left| \sum_{n=N}^\infty\!a_n \right| < M$ for any $\displaystyle N \geq 1$
My assumtition must be wrong.
Either does both diverge to $\displaystyle \pm\infty$ or converge.

Is there something wrong? Something missing to complete the proof?

2. Originally Posted by fgn

Is there something wrong? Something missing to complete the proof?
I did not work on the proof but my trianed eagle eye konstantly catches mistakes. You approached this by contradiction. You said "Assume this does not converge...." Then it diverges to $\displaystyle \pm \infty$. That is not true. The sequence $\displaystyle \{(-1)^n\}$ does not converge neither does it diverge to $\displaystyle \pm \infty$

3. Originally Posted by ThePerfectHacker
The sequence $\displaystyle \{(-1)^n\}$ does not converge neither does it diverge to $\displaystyle \pm \infty$
You're right. That case botherd me a bit..
But if I divided it into two cases, one case that diverge to $\displaystyle \pm\infty$ and one diverge like your example. I.e the partial sum is oscillating between some value. Then it should be trival to show that your example diverge for any starting index.

4. There is really no need for cases.
Remember that series convergence is about convergence of sequences of partial sums. There is a nice results about sequences: If $\displaystyle \left( {c_n } \right)$ is a sequence and r is number then $\displaystyle \left( {c_n } \right)$ converges if and only if the sequence $\displaystyle \left( {r + c_n } \right)$ converges.

Realizing that $\displaystyle \pm \sum\limits_{n = 1}^N {a_n }$ is a number the result follows quickly.

This result is usually summarized by the old saying: Series convergence is completely determined by what happens to its tail.