Solved on Nov 27, 2023

Find the center of the power series xn/n3\sum x^n / n^3. The options are: a) 2, b) 3, c) 1/21/2, d) -1, e) 1/2-1/2, f) -1, g) 1, h) -3, i) 0.

STEP 1

Assumptions1. The power series is given by xn/n3\sum x^{\wedge} n / n^{\wedge}3. . We are asked to find the center of this power series.

STEP 2

The center of a power series is the value of xx where the series is centered. In other words, it's the value of xx that makes the series converge to a finite value.

STEP 3

For a power series of the form an(xc)n\sum a_n (x - c)^n, the center of the series is cc.

STEP 4

In our case, the power series is given by xn/n3\sum x^{\wedge} n / n^{\wedge}3. This can be rewritten as (x0)n/n3\sum (x -0)^n / n^{\wedge}3.

STEP 5

Comparing this with the general form of the power series, we can see that the center cc is0.
Therefore, the center of the power series xn/n3\sum x^{\wedge} n / n^{\wedge}3 is0.

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