How do you find the power series for #f'(x)# and #int_0^x f(t)dt# given the function #f(x)=sum_(n=0)^oo x^(2n)# ?

Question

Ezekiel Alexander · Accepted Answer

# f'(x) = 0 + 2x^1 + 4x^3 + 6x^5 + ... #
# " " = sum_(n=1)^oo (2n)x^(2n-1) #
# int_0^x \ f(t) \ dt = x + x^3/3 + x^5/5 + x^7/7 + ... #
# " " = sum_(n=0)^oo x^(2n+1)/(2n+1) #

We have #f(x)# defined by its power series: # f(x) = sum_(n=0)^oo x^(2n) # We can expand the fist few terms to get an idea of how the series behaves: # f(x) = x^0 + x^2 + x^4 + x^6 + ... # # " " = 1 + x^2 + x^4 + x^6 + ... # So first we find the derivative by differentiating term by term: # f'(x) = 0 + 2x^1 + 4x^3 + 6x^5 + ... # # " " = sum_(n=1)^oo (2n)x^(2n-1) # And similarly we find the integral by integrating term by term: # int_0^x \ f(t) \ dt = int_0^x \ 1 + t^2 + t^4 + t^6 + ... \ dt# # " " = [t + t^3/3 + t^5/5 + t^7/7 + ... ]_0^x # # " " = {x + x^3/3 + x^5/5 + x^7/7 + ... } - {0} # # " " = x + x^3/3 + x^5/5 + x^7/7 + ... # # " " = sum_(n=0)^oo x^(2n+1)/(2n+1) #

Christopher Agresta · Answer

undefined To find the power series for $ f'(x) $ and $ \int_{0}^{x} f(t) dt $ given the function $ f(x) = \sum_{n=0}^{\infty} x^{2n} $, we first need to find the derivatives of $ f(x) $ and integrate it with respect to $ t $.

Starting with $ f(x) $:
$$ f(x) = \sum_{n=0}^{\infty} x^{2n} $$

To find $ f'(x) $, we differentiate each term of the series with respect to $ x $:
$$ f'(x) = \sum_{n=0}^{\infty} \frac{d}{dx} (x^{2n}) $$

$$ f'(x) = \sum_{n=0}^{\infty} 2nx^{2n-1} $$

Now, for $ \int_{0}^{x} f(t) dt $, we integrate each term of $ f(x) $ with respect to $ t $:
$$ \int_{0}^{x} f(t) dt = \int_{0}^{x} \sum_{n=0}^{\infty} t^{2n} dt $$

$$ \int_{0}^{x} f(t) dt = \sum_{n=0}^{\infty} \int_{0}^{x} t^{2n} dt $$

$$ \int_{0}^{x} f(t) dt = \sum_{n=0}^{\infty} \frac{t^{2n+1}}{2n+1} \bigg|_{0}^{x} $$

$$ \int_{0}^{x} f(t) dt = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{2n+1} $$

Emma Aldridge · Answer

undefined To find the power series for $ f'(x) $, differentiate the given function term by term to obtain each term of the derivative. Then, write the resulting series.

For the integral $ \int_{0}^{x} f(t) \, dt $, integrate the given function term by term to obtain each term of the integral. Then, write the resulting series.

Since the given function is a power series, its derivative and integral can be computed by differentiating and integrating each term of the series individually.