Differentiate the given series expansion of f  term-by-term to obtain the corresponding series expansion for the derivative of f . if f(x)=(1)/((1-x^(5)))=\sum_(n=0)^(\infty ) x^(5n)  f'(x) = \sum_(n=1)^(\infty ) 

To find the series expansion for the derivative of the given function \( f(x) \), we will differentiate the given series expansion of \( f(x) \) term-by-term. The given series expansion is:

\[ f(x) = \sum_{n=0}^{\infty} x^{5n} \]

We need to differentiate this series term-by-term to find the series expansion for \( f'(x) \):

\[ f'(x) = \frac{d}{dx} \left( \sum_{n=0}^{\infty} x^{5n} \right) \]

Using the linearity of differentiation and the fact that we can interchange the summation and differentiation (since the series is uniformly convergent), we get:

\[ f'(x) = \sum_{n=0}^{\infty} \frac{d}{dx} \left( x^{5n} \right) \]

To differentiate each term, we use the power rule of differentiation:

\[ \frac{d}{dx} \left( x^{5n} \right) = 5n x^{5n-1} \]

Therefore, we have:

\[ f'(x) = \sum_{n=0}^{\infty} 5n x^{5n-1} \]

Since the term for \( n = 0 \) is zero (because \( 5 \cdot 0 \cdot x^{-1} = 0 \)), we can start the summation from \( n = 1 \):

\[ f'(x) = \sum_{n=1}^{\infty} 5n x^{5n-1} \]

Thus, the series expansion for the derivative of \( f(x) \) is:

\[ \boxed{f'(x) = \sum_{n=1}^{\infty} 5n x^{5n-1}} \]

Do you want details or have any questions?

Here are 8 related questions that expand on the previous information:

1. How can we verify the convergence of the differentiated series?
2. What is the radius of convergence for the series expansion of \( f'(x) \)?
3. How would the differentiation change if the series started at \( n = 1 \) instead of \( n = 0 \)?
4. Can the method of term-by-term differentiation be applied to non-uniformly convergent series?
5. What are the conditions under which term-by-term differentiation is valid?
6. How does the derivative of the series expansion compare with the original function \( f(x) \)?
7. What is the significance of the power series in this context?
8. How can we express the derivative \( f'(x) \) in a closed form if possible?

**Tip:** When differentiating a power series term-by-term, ensure that the series converges uniformly on the interval where you differentiate. This guarantees the validity of interchanging the summation and differentiation operations.

Learn how to differentiate the series expansion of f(x)=(1)/(1-x^5) term-by-term to find the series expansion for f'(x). Understand the application of the power rule of differentiation in power series and series expansion. Suitable for undergraduate-level mathematics.

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