# 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)# ?

# 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 can expand the fist few terms to get an idea of how the series behaves:

So first we find the derivative by differentiating term by term:

And similarly we find the integral by integrating term by term:

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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} ]

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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.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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