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Moderate

Shifting a Power Series Base Point I

SVCALC-WYPNLX

If the power series $x+x^2+x^3+\ldots$ is written as a power series at $\frac{1}{3}$:

$$x+x^2+x^3+\ldots=c_0+c_1\left(x-\frac{1}{3}\right)+c_2\left(x-\frac{1}{3}\right)^2+\ldots$$

...what is the value of the coefficient $c_0$?

A

$0$

B

$\cfrac{1}{2}$

C

$\cfrac{2}{3}$

D

$\cfrac{3}{4}$

E

$\cfrac{3}{2}$