((5n^3)/(2^n))(x-1)^n
What is the center of this power series? (in either words, what is value of a when this power series is about a)
In order to find the center of the power series, we need to examine the variable "x" in the expression. The power series is given by ((5n^3)/(2^n))(x - 1)^n.
The center of a power series is the value of "x" around which the power series is expanded. In general, the center is denoted by "a." In this case, we are looking for the value of "a" that makes the power series valid.
To find the center, we need to analyze the term (x - 1)^n. When expanding this expression using the binomial theorem, it becomes a series of terms of the form nCk * x^(n-k) * (-1)^k.
Notice that in this term, the variable "x" is raised to the power of (n-k). In order for this series to be valid, (n-k) should always be a non-negative integer. This means that the terms are only valid when (n-k) is greater than or equal to zero.
Since we don't have any restriction for "n," we need to find a value for "a" such that (a - 1) raised to any positive power "n" will always be valid. Therefore, the center of this power series is x = 1.
In conclusion, the center of this power series is x = 1.