Question

Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.

g(x) = − 1/(1 + x)² using f(x) = 1/(1 + x)

Accepted Answer

Recall the given function and its power series: we have $$ f(x) = \frac{1}{1+x} . $$The power series representation of $$ f(x) $$ centered at 0 is the geometric series $$ \sum_{n=0}^\infty (-1)^n x^n $$, valid for $$ |x| \u003c 1 . $$Note that $$ g(x) = -\frac{1}{(1+x)^2} $$ can be expressed in terms of the derivative of $$ f(x) . $$Specifically, differentiate $$ f(x) $$ with respect to $$ x $$: $$ f'(x) = \frac{d}{dx} \left( \frac{1}{1+x} \right) = -\frac{1}{(1+x)^2} . $$This means $$ g(x) = -f'(x) . $$Differentiate the power series for $$ f(x) $$ term-by-term: $$ \frac{d}{dx} \sum_{n=0}^\infty (-1)^n x^n = \sum_{n=0}^\infty (-1)^n \frac{d}{dx} x^n = \sum_{n=1}^\infty (-1)^n n x^{n-1} . $$Note that the $$ n=0 $$ term vanishes upon differentiation. Since $$ g(x) = -f'(x) $$, multiply the differentiated series by $$ -1  to $$get the power series for $$ g(x) $$: $$ g(x) = - \sum_{n=1}^\infty (-1)^n n x^{n-1} = \sum_{n=1}^\infty (-1)^{n+1} n x^{n-1} . $$The interval of convergence remains the same as for $$ f(x) $$ because differentiation does not change the radius of convergence. Therefore, the interval of convergence for $$ g(x)  is$$ $$ |x| \u003c 1 .$$

Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.

g(x) = − 1/(1 + x)² using f(x) = 1/(1 + x)

Verified video answer for a similar problem:

Key Concepts

Power Series Representation

Term-by-Term Differentiation and Integration of Power Series

Interval of Convergence

Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.g(x) = − 1/(1 + x)² using f(x) = 1/(1 + x)

Verified video answer for a similar problem:

Key Concepts

Power Series Representation

Term-by-Term Differentiation and Integration of Power Series

Interval of Convergence

Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.

g(x) = − 1/(1 + x)² using f(x) = 1/(1 + x)