# How do you find the power series for #f'(x)# and #int f(t)dt# from [0,x] given the function #f(x)=Sigma n^-3x^n# from #n=[1,oo)#?

For

We start determining the radius of convergence of the series:

using the ratio test:

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To find the power series for ( f'(x) ) and ( \int f(t) , dt ) from ( [0,x] ) given the function ( f(x) = \sum_{n=1}^{\infty} n^{-3}x^n ) from ( n=1 ) to ( \infty ), you first need to differentiate and integrate the given power series term by term.

For ( f'(x) ), differentiate each term of the series with respect to ( x ), then express the resulting series as a new power series. For ( \int f(t) , dt ), integrate each term of the series with respect to ( t ), then express the resulting series as another power series.

After finding the power series representations for ( f'(x) ) and ( \int f(t) , dt ), you can manipulate them further as needed.

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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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