If f(x)=(1-x)^-1,|x|<1 then the value of (f^'i(0))/(f(0))+(f^i i '(0))/(f^i(0))+(f^i r(0))/(f^i

step 1 The function effect is given to be 1 minus x x for absolute value of x less than 1. So on sex we

If f(x)=(1-x)^-1,|x|<1 then the value of...

If $f(x)=(1-x)^{-1},|x|<1$ then the value of $\frac{f^{\prime i}(0)}{f(0)}+\frac{f^{i i \prime}(0)}{f^{i}(0)}+\frac{f^{i r}(0)}{f^{i i}(0)}+\ldots+\frac{f^{n+1}(0)}{f^{n-1}(0)}$ is (a) $\frac{n(n+1)(n+2)}{3}$ (b) $\frac{n(n+1)(2 n+1)}{6}$ (c) $\frac{n(n+1)(n+2)}{6}$ (d) None of the above

If $f(x)=(1-x)^{-1},|x|<1$ then the value of $\frac{f^{\prime i}(0)}{f(0)}+\frac{f^{i i \prime}(0)}{f^{i}(0)}+\frac{f^{i r}(0)}{f^{i i}(0)}+\ldots+\frac{f^{n+1}(0)}{f^{n-1}(0)}$ is
(a) $\frac{n(n+1)(n+2)}{3}$
(b) $\frac{n(n+1)(2 n+1)}{6}$
(c) $\frac{n(n+1)(n+2)}{6}$
(d) None of the above

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Maclaurin Series

A Maclaurin series is the expansion of a function into a power series around 0. It expresses the function as an infinite sum using its derivatives at 0. In many problems, especially with functions like (1 - x)?¹, this expansion simplifies computation of derivatives and evaluation at specific points.

Differentiation of Power Series

When a function is expressed as a power series, its derivatives can be calculated term?by-term. Since the coefficients often involve factorials, each successive derivative follows a clear, predictable pattern. This property is key to understanding ratios of successive derivatives evaluated at 0.

Factorial Properties

Factorials arise naturally in the context of power series and derivatives. They often simplify the ratio of a (k+1)th derivative to the kth derivative, as the factorial terms cancel in a predictable manner, leading to simple integer sequences.

Arithmetic Series

An arithmetic series is a sum of a sequence of numbers in which each term increases by a constant difference. Commonly derived formulas for the sum of arithmetic series provide a quick and effective method to evaluate sums that appear when successive integer ratios are summed, as is the case when the simplified derivative ratios create an arithmetic sequence.

Transcript

Please give Ace some feedback