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When you multiply a series by x, therefore changing the nth power on x, is it mandatory that you change the number where you start counting? This is confusing me because some problems do both, where others explain it in two different separate steps.

Basically,

If I am multiplying a taylor series by x, then am I supposed to add one to the index AND count one lower? or is it only mandatory to change the power on x, not the counting number?

If the index [nth power on x] is changed, is it mandatory to change where the series starts counting at?

I keep seeing problems where they only change one, and not the others. I thought you're supposed to replace all n's by the new relation.

I hope this makes sense, lol.