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!