91Ó°ÊÓ

A power series is an infinite series of the form \(\sum_{k=0}^{\infty}c_k(x-a)^k\). Comparing this definition to the given expression, \(\sum_{i=0}^{\infty}(5x-20)^{k}\), we can see that it indeed has the form of a power series.

We can determine the center of the power series by comparing the given expression with the standard form of a power series, \(\sum_{k=0}^{\infty}c_k(x-a)^k\). In our case, the given expression is \((5x-20)^k\), and the corresponding general term in the standard power series is \((x-a)^k\). Equating the two expressions, we have: \((5x-20)^k = (x-a)^k\) Since the exponents are equal, we can simplify the equation by comparing the bases: \(5x-20 = x-a\) Now, we solve for \(a\): \(a = 5x-x-20\) \(a= 4x-20\) Since the center \(a\) should be a constant, we solve for \(x\) when \(a = 4x - 20\): \(4x - 20 = a\) Adding 20 to both sides: \(4x = a+20\) Dividing by 4: \(x = \frac{a + 20}{4}\) Now, to find the value of \(a\) that makes the right side a constant, we set the numerator equal to zero: \(a + 20 = 0\) Solving for \(a\): \(a = -20\) So, the center of the power series is \(a = -20\).

Now, we know that \((5x-20)^k =(x-a)^k = (x - (-20))^k = (x + 20)^k\). Thus, we can see that the coefficients \(c_k\) in our given expression are all equal to 1, as there is no additional multiplier in the expression with the power of \(k\). Therefore, the formula for the coefficients is simply: \(c_k = 1\) for all values of \(k\). #Conclusion# In conclusion, the given expression \(\sum_{i=0}^{\infty}(5x-20)^{k}\) is a power series with a center at \(a = -20\) and coefficients of \(c_k = 1\) for all values of \(k\).