Final answer:
To expand the function 65− in a power series with center 0, we can use the equation 11−=∑=0[infinity] for ||<1. Let's plug in the function 65− into the power series equation and find the coefficients using the binomial theorem.
Step-by-step explanation:
To expand the function 65− in a power series with center 0, we can use the equation 11−=∑=0[∞] for ||<1. This equation represents a power series, where we sum the terms from n=0 to infinity. Let's plug in the function 65− into the power series equation:
65− = ∑(n=0 to infinity) a_n(x-0)^n
To find the coefficients, we can use the binomial theorem, which states that (a+b)^n = ∑(k=0 to n) (n choose k) a^(n-k)b^k. Let's substitute a = 6, b = -21, and x = [(6−21x)] into the binomial theorem equation:
(6-21x)^5 = ∑(k=0 to 5) (5 choose k) 6^(5-k)(-21)^k
Expanding this equation will give us the power series expansion of 65− with center 0.