Taylor Series Multiplication: Why is the Exponent Not Doubled?

[asdf1]

for a termwise mulitiplication of two taylor series,
1.sigma a m(x-x0)^m
2.sigma b m (x-x0)^m
then if you mutilpy 1 and 2, shouldn't you get
sigma a mbm(x-x0)^2m?
but my textbook says that it's only
sigma a mbm(x-x0)^m

from advanced engineering mathematics by erwin kreyszig 9E pg202

 

[EnumaElish]

For termwise multiplication $a_m b_m (x - x_0)^{2m}$ is right.

If you were adding the two together then $(a_m + b_m)(x - x_0)^{m}$ would have been right.

On the other hand, it just may be possible that your textbook defines termwise Taylor series multiplication as $a_m b_m (x - x_0)^{m}$. In which case it would be right on its own accord.

 

[matt grime]

It would certainly appear to be the definition that you're having trouble with. Ie, that is the definition of termwise mult. I can see why it is morally wrong to think it a_mb_m(x-x_0)^2m

since you should be thinking of the x-x_0 part merely as a place holder.

Thus a taylor series is really a sequence (a_m) m in N, and addition and multiplication of the series is done as for the sequences.

 

[asdf1]

thank you very much! :)