Is this a misprint?

Question

I found this in a mathematics pdf, I was wondering if I'm right about a mistake.

Shouldn't the last summation be from $m=1$ to $n-1$? Thanks.

EDIT: $T_m$ is 1 for even $m$, 0 for odd $m$.

Answer 1

We can find directly that:

$$a_{0}+\sum_{m=1}^{n}T_{m}a_{m}-\sum_{m=0}^{n-1}T_{m}a_{m+1}=a_{0}+T_{n}a_{n}+\sum_{m=1}^{n-1}T_{m}a_{m}-\sum_{m=1}^{n-1}T_{m}a_{m+1}-T_{0}a_{1}\\ =a_{0}+T_{n}a_{n}+\sum_{m=1}^{n-1}T_{m}(a_{m}-a_{m+1})-T_{0}a_{1}\\ =T_{n}a_{n}+(a_{0}-T_{0}a_{1})+\sum_{m=1}^{n-1}T_{m}(a_{m}-a_{m+1}).$$

The last line above can be written as $T_{n}a_{n}+\sum_{m=0}^{n-1}T_{m}(a_{m}-a_{m+1})$ if and only if we can write $a_{0}-T_{0}a_{1}$ as $T_{0}(a_{0}-a_{1})$.

$$a_{0}-T_{0}a_{1}=T_{0}(a_{0}-a_{1}) \iff a_{0}=T_{0}a_{0} \iff T_0=1 \lor a_0=0.$$

So assuming either $T_0=1$ or $a_0=0$, the last summation is correct as printed.