https://file.aiquickdraw.com/mathbot/file/172555654122422dtzpcn.jpg

The mathematical expression from the image appears to be:

\[
\lim_{x \to 0} \left( \sum_{n=1}^{\infty} \frac{n^3}{3^n} e^{3x} \sin(nx) \right)
\]

To solve this, let's break it down step by step:

### Step 1: Separate the summation and the exponential part
The summation involves terms of the form:
\[
\frac{n^3}{3^n} e^{3x} \sin(nx)
\]
We are taking the limit as \( x \to 0 \), so we will evaluate the behavior of this function as \( x \to 0 \).

### Step 2: Apply the limit to each term
Since \( e^{3x} \to 1 \) as \( x \to 0 \), we can simplify the expression:
\[
\lim_{x \to 0} \left( \sum_{n=1}^{\infty} \frac{n^3}{3^n} \sin(nx) \right)
\]

### Step 3: Analyze the behavior of \( \sin(nx) \)
For small values of \( x \), the sine function can be approximated using its Taylor series expansion:
\[
\sin(nx) \approx nx \quad \text{as} \quad x \to 0
\]
Thus, the expression becomes:
\[
\lim_{x \to 0} \left( \sum_{n=1}^{\infty} \frac{n^3}{3^n} nx \right) = x \lim_{x \to 0} \left( \sum_{n=1}^{\infty} \frac{n^4}{3^n} \right)
\]

### Step 4: Simplify the summation
Now we need to compute the summation:
\[
\sum_{n=1}^{\infty} \frac{n^4}{3^n}
\]
This is a standard power series, and the result of such a series can be obtained using special techniques or reference to known summation results.

### Step 5: Conclusion
The limit \( x \to 0 \) makes the entire expression \( x \times \) a constant (from the sum), which evaluates to 0 because of the multiplying factor of \( x \). Thus:
\[
\lim_{x \to 0} \left( \sum_{n=1}^{\infty} \frac{n^3}{3^n} e^{3x} \sin(nx) \right) = 0
\]

### Final Answer:
The limit is:
\[
0
\]

---

Would you like further details on how the summation was handled or more explanation on the steps? Here are five related questions:

1. How do you calculate power series involving exponential functions?
2. What is the Taylor series expansion of \( \sin(x) \) and how is it used in limits?
3. How do limits affect infinite sums in calculus?
4. What techniques can be used to simplify complicated sums like \( \sum \frac{n^4}{3^n} \)?
5. How does the small angle approximation of trigonometric functions work?

**Tip:** Always look for ways to approximate functions, like using Taylor series, when dealing with limits. It simplifies complex expressions!

Evaluate the limit as x approaches 0 for the series sum from n equals 1 to infinity of (n^3 / 3^n) * e^(3x) * sin(nx).

Learn how to evaluate the limit of a series involving exponential and trigonometric functions as x approaches 0. This problem covers key concepts such as limits, series, exponential functions, and the sine function, with step-by-step solutions. Suitable for undergraduate students.

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