If I have a power series $$\sum _{k}^{\infty }f(x)$$ and I differentiate it I get according to my current knowledge $\sum _{k}^{\infty }f(x)'$,however when I look at a power series defined by $$\sum _{0}^{\infty }a_{n}\left( x-c\right) ^{n}$$ When one takes the derivative you get$$\sum _{1}^{\infty }n\cdot a_{n}\left( x-c\right) ^{n-1}$$Why is this the case? Does this hold for general series? Did I miss something obvious? When taking derivative at specific point does one just plug in the value of x in $f(x)'$ inside the $\sum $? Does such a operation even exist? Does the derivative operation when applied to a series have a physical interpretation in the real world? Does the definite for series follow from this definition? I sincerely apologize if I asked to many questions.
2026-04-06 03:38:49.1775446729
Differentiation and integration of a series
78 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in SEQUENCES-AND-SERIES
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