Use the Binomial Series to Expand the Function as a Power Series 41x

To expand the function 41x using the binomial series, we start by expressing it in a form that can be applied to the binomial theorem.

The binomial series states that for any real number α, we have:

(1 + x)α = Σ (α choose k) x^k (for |x| < 1)

In this case, we want to expand 41x. First, we can rewrite this as:

41x = 41(1 + (x – 1))

Now we note that if we let x be small enough (for example, if we’re looking around the point x = 1), we can set up our series expansion using the binomial theorem. The series expansion for (1 + u) where u = (x – 1) is given by:

Σ (k choose n) (x – 1)ⁿ (for n=0 to ∞)

Thus, substituting back, we get the expansion as:

41(1 + (x – 1) + (x – 1)²/2! + (x – 1)³/3! + … )

By multiplying through by 41, we derive:

41 + 41(x – 1) + 41(x – 1)²/2! + 41(x – 1)³/3! + …

In conclusion, using the binomial series, we have expanded 41x as a power series around the point x = 1.