In this site, on proof 1 section there is a line where the expression is split into even and odd $n$, I don't understand how this was done, could someone explain it for me?

  • $\begingroup$ Write out the entire sum, then pretend each pair of terms is one term of another sum. $\endgroup$ – Element118 Nov 29 '15 at 1:40

What it did was that it changed this original sequence $$\sum_{m=0}^{\infty}a_m=a_0+a_1+a_2+...$$ into $$\sum_{n=0}^{\infty}(a_{2n}+a_{2n+1})=(a_0+a_1)+(a_2+a_3)+...$$ by showing two adjacent terms at the same time.

So $n=0$ in the new sequence would correspond to $m=0$ & $1$ in the old sequence, and $n=1$ corresponds to $m=2$ & $3$, and $n=2$ to $m=4$ & $5$, and so on.

This way, $(-1)^m$ can be split into two cases - $m$ is even and $m$ is odd, and can be explicitly calculated.

  • $\begingroup$ Thank you. I didn't think it was so simple $\endgroup$ – João Nov 29 '15 at 13:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.