sin 2x, sin²x sin 2x का nवां अवकलन गुणांक ज्ञात कीजिए। Test the series whose n^{th} term is Σ x^{n}/n! Prove that :
Understand the Problem
प्रश्न में हमें एक श्रृंखला का परीक्षण करने के लिए कहा गया है, जिसका nवां पद समान्य सूत्र के रूप में दिया गया है। यह सवाल हमें यह साबित करने के लिए कहता है कि इसे किस प्रकार परखा जा सकता है।
Answer
The nth derivative of $f(x) = \sin^2(x) \sin(2x)$ can be given as a formula through repeated differentiation, and the series $\sum \frac{x^n}{n!}$ converges to $e^x$.
Answer for screen readers
The nth derivative of $f(x) = \sin^2(x) \sin(2x)$ is given by a specific formula which can be derived through repeated differentiation. The series $\sum \frac{x^n}{n!}$ converges to $e^x$.
Steps to Solve
- Find the nth Derivative of the Function
We need to find the nth derivative of the function $f(x) = \sin^2(x) \sin(2x)$.
- Differentiate Using Product Rule
Using the product rule for derivatives, we differentiate: $$ f'(x) = \frac{d}{dx}(\sin^2(x)) \cdot \sin(2x) + \sin^2(x) \cdot \frac{d}{dx}(\sin(2x)) $$
- Applying Chain and Product Rules
- For the first term, use the chain rule: $$ \frac{d}{dx}(\sin^2(x)) = 2\sin(x)\cos(x) = \sin(2x) $$
- For the second term: $$ \frac{d}{dx}(\sin(2x)) = 2\cos(2x) $$
This results in: $$ f'(x) = \sin(2x)\sin(2x) + 2\sin^2(x)\cos(2x) $$
- Simplifying the Derivative
Express $f'(x)$ in manageable form to find higher derivatives. Continue this process to find $f^{(n)}(x)$ efficiently.
- Confirming the Series
For the series whose nth term is $\frac{x^n}{n!}$, we acknowledge it represents the Taylor series expansion for $e^x$. We need to prove its convergence for all $x$.
The nth derivative of $f(x) = \sin^2(x) \sin(2x)$ is given by a specific formula which can be derived through repeated differentiation. The series $\sum \frac{x^n}{n!}$ converges to $e^x$.
More Information
The function $\sin^2(x) \sin(2x)$ combines trigonometric identities, and its derivatives encapsulate oscillatory behavior. The series $\sum \frac{x^n}{n!}$ converges for all x, connecting to the exponential function.
Tips
- Forgetting to apply the product rule correctly for the derivatives leads to miscalculated terms.
- Misapplying trigonometric identities can complicate finding the nth derivative.
- Not recognizing the convergence of the series can lead to invalid conclusions about its behavior.
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