Math Problem Statement
Solution
The problem requires using the ratio test to check the convergence of the series:
Solution
To apply the ratio test, we consider the terms of the series:
The ratio test states that a series converges absolutely if
Step 1: Compute
-
For :
-
Find :
Step 2: Analyze
Notice that oscillates depending on the value of :
- When , .
- When , .
- For even values of , , which implies that some terms of the series are zero.
Step 3: Take the Limit as
Since oscillates and does not affect the factorial in the denominator, we focus on .
Thus, we get:
Conclusion
Since the limit is , the series converges absolutely by the ratio test.
Would you like further details or have any questions?
Related Questions
- How would the convergence be affected if the factorial term were removed from the denominator?
- Can the ratio test be used for all types of series?
- What other convergence tests could be applied to this series?
- How does the oscillating behavior of impact the series' convergence?
- What would happen if we replaced with another trigonometric function?
Tip: The ratio test is especially useful when terms involve factorials or exponential functions, as it simplifies the comparison of successive terms.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Series Convergence
Ratio Test
Trigonometric Series
Factorials
Formulas
Ratio Test for series convergence
Factorial n!
Theorems
Ratio Test
Suitable Grade Level
College/University Level (Calculus)
Related Recommendation
Ratio Test for Convergence of the Series with Sine and Factorial Terms
Determine Convergence of ∑ n * sin(π / 2^n) Using D'Alembert's Criterion
Convergence Test of Series \( \sum \frac{\sin(n)}{n!} \) Using the Ratio Test
D'Alembert Ratio Test for Series Convergence with Factorials
Determining Convergence of a Series Using the Ratio Test with Factorials