To find the series expansion for the indefinite integral of the given function $f(x)$, we will integrate the given series expansion of $f(x)$ term-by-term. The given series expansion is:

$f(x) = \sum_{n=0}^{\infty} (-1)^n 7n x^{7n-1}$

We need to integrate this series term-by-term from 0 to $x$ to find the series expansion for $\int_0^x f(t) \, dt$:

$\int_0^x f(t) \, dt = \int_0^x \left( \sum_{n=0}^{\infty} (-1)^n 7n t^{7n-1} \right) dt$

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

$\int_0^x f(t) \, dt = \sum_{n=0}^{\infty} (-1)^n 7n \int_0^x t^{7n-1} \, dt$

To integrate each term, we use the power rule of integration:

$\int t^{7n-1} \, dt = \frac{t^{7n}}{7n}$

Therefore, we have:

$\int_0^x t^{7n-1} \, dt = \left[ \frac{t^{7n}}{7n} \right]_0^x = \frac{x^{7n}}{7n}$

Substituting this back into our series, we obtain:

$\int_0^x f(t) \, dt = \sum_{n=0}^{\infty} (-1)^n 7n \cdot \frac{x^{7n}}{7n} = \sum_{n=0}^{\infty} (-1)^n x^{7n}$

However, note that the series starts from $n = 1$ to ensure the correct form based on the given problem:

$\int_0^x f(t) \, dt = \sum_{n=1}^{\infty} (-1)^n x^{7n}$

Thus, the series expansion for the indefinite integral of $f(x)$ is:

$\boxed{\int_0^x f(t) \, dt = \sum_{n=1}^{\infty} (-1)^n x^{7n}}$

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 given series expansion?
2. What is the radius of convergence for the series expansion of $f(x)$?
3. How would the integration change if the series started at $n = 1$ instead of $n = 0$?
4. Can the method of term-by-term integration be applied to non-uniformly convergent series?
5. What are the conditions under which term-by-term integration is valid?
6. How does the integral of the series expansion compare with the original function $f(x)$?
7. What is the significance of the alternating series in this context?
8. How can we express the integral $\int_0^x f(t) \, dt$ in a closed form if possible?

Tip: When integrating a power series term-by-term, ensure that the series converges uniformly on the interval of integration. This guarantees the validity of interchanging the summation and integration operations.