Let's solve this step-by-step.

Given Function:

$f(x) = 7 e^{-4x}$

The Maclaurin series is the Taylor series expanded around $x = 0$. For $f(x) = 7 e^{-4x}$, we first expand $e^{-4x}$ using its Maclaurin series:
$e^{-4x} = \sum_{n=0}^\infty \frac{(-4x)^n}{n!} = \sum_{n=0}^\infty \frac{(-1)^n (4^n x^n)}{n!}.$
Multiplying by 7 gives:
$f(x) = 7 \sum_{n=0}^\infty \frac{(-1)^n (4^n x^n)}{n!}.$

Now we find the first four nonzero terms.

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(a) First Four Nonzero Terms

To compute the first four terms, plug in $n = 0, 1, 2, 3$:

1. First term ($n = 0$):
   $f(x) = 7 \cdot \frac{(-1)^0 (4^0 x^0)}{0!} = 7.$

2. Second term ($n = 1$):
   $f(x) = 7 \cdot \frac{(-1)^1 (4^1 x^1)}{1!} = 7 \cdot (-4x) = -28x.$

3. Third term ($n = 2$):
   $f(x) = 7 \cdot \frac{(-1)^2 (4^2 x^2)}{2!} = 7 \cdot \frac{16x^2}{2} = 7 \cdot 8x^2 = 56x^2.$

4. Fourth term ($n = 3$):
   $f(x) = 7 \cdot \frac{(-1)^3 (4^3 x^3)}{3!} = 7 \cdot \frac{-64x^3}{6} = 7 \cdot \left(-\frac{64}{6}x^3\right) = -\frac{224}{3}x^3.$

Thus, the first four nonzero terms are:
$7 - 28x + 56x^2 - \frac{224}{3}x^3.$

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(b) Power Series in Summation Notation

The general term for the series is:
$a_n = 7 \cdot \frac{(-1)^n (4^n)}{n!} x^n.$

Thus, the power series is:
$f(x) = \sum_{n=0}^\infty 7 \cdot \frac{(-1)^n (4^n)}{n!} x^n = \sum_{n=0}^\infty \frac{7(-1)^n (4^n)}{n!} x^n.$

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(c) Interval of Convergence

To determine the interval of convergence, we analyze the original series:
$\sum_{n=0}^\infty \frac{7(-1)^n (4^n)}{n!} x^n.$

Here, the factorial $n!$ in the denominator ensures that the series converges for all $x$ (since the exponential series always converges for all $x$).

Conclusion: The interval of convergence is:
$(-\infty, \infty).$

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Final Answers

(a) First four nonzero terms:
$7 - 28x + 56x^2 - \frac{224}{3}x^3.$

(b) Power series in summation notation:
$f(x) = \sum_{n=0}^\infty \frac{7(-1)^n (4^n)}{n!} x^n.$

(c) Interval of convergence:
$(-\infty, \infty).$

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Would you like a detailed explanation of any part of this process?

Here are some related questions you might want to explore:

1. What is the general method for finding a Maclaurin series?
2. How does factorial growth affect the convergence of a power series?
3. What are the properties of the exponential function's series expansion?
4. How would the series change if the coefficient $7$ were replaced by another constant?
5. How do we test for convergence in a general series?

Tip: When working with power series, always verify convergence using the ratio test or by analyzing the growth of terms directly!