Math 2414 Series Project Due

Math 2414 Series Project Due

Find the sum of the series. Find the interval and radius of convergence for the following series: 7. 1 0 3 k k k k x 8. ) ! k k k x k 9. 1 4 k k k x k 10. 0 !( 5) k k k x 11. k k k k x 12. Construct the Taylor series expansion for 2 ( ) x f x e , about 0x = . 13. Construct the Taylor series expansion for ( ) sin( )f x x , about 3 2x = . 14. From the list of basic power series, ̊

Paper For Above instruction

The series and convergence problems provided in this assignment encompass fundamental concepts in infinite series and power series, essential topics in advanced calculus. Understanding how to evaluate the sum of series, determine their convergence behavior, and develop power series expansions including Taylor series, allows mathematicians and students to analyze functions and approximation techniques effectively.

Sum of the Series and Convergence Analysis

The initial task involves calculating the sum of a specified infinite series. Although the specific series expressions are not numerically provided in the prompt, typical examples include geometric series or other common types. For instance, the sum of a geometric series with first term \( a \) and common ratio \( r \), such that \( |r|

The subsequent problems require determining the radius and interval of convergence for various power series. These series are generally of the form \(\sum_{k=0}^\infty a_k x^k\), where the radius of convergence \( R \) can be found using the Ratio Test or Root Test. For the series in problem 7, which appears to involve a general \( k \) with \( x^k \) terms, the ratio \( \lim_{k \to \infty} |\frac{a_{k+1}}{a_k}| \) is instrumental in establishing \( R \), and the interval is then found by testing endpoint convergence.

Constructing Taylor Series Expansions

Problems 12 and 13 focus on deriving Taylor series expansions for exponential and sine functions centered at specific points. The Taylor series for \( e^x \) around zero is well-known:

\[

e^x = \sum_{k=0}^\infty \frac{x^k}{k!}

\]

which converges for all real \( x \). To create the Taylor series for \( e^x \) about \( x=0 \), this expansion is directly applicable.

Similarly, the Taylor series for \( \sin x \) centered at \( x=3/2 \) involves shifting the expansion:

\[

\sin x = \sum_{k=0}^\infty (-1)^k \frac{(x - 3/2)^{2k+1}}{(2k+1)!}

\]

which converges for all \( x \). Developing these series involves calculating derivatives at the center point and applying the Taylor formula.

Power Series for Logarithmic Functions

The problem prompts the creation of a power series for \( \ln(1+x) \). From the known basic series:

\[

\ln(1+x) = \sum_{k=1}^\infty (-1)^{k+1} \frac{x^k}{k}

\]

valid for \( -1

Using this, one can derive a power series for \( \ln(1+x) \), analyze its domain, and interpret it for various applications, including the evaluation of integrals involving logarithmic functions. Applying term-by-term integration, the series can be integrated to find the power series representation of the definite integral:

\[

\int \ln(1+x) \, dx

\]

which involves integrating each term of the series and analyzing the convergence within its domain.

Conclusion

This assignment integrates core concepts in series and power series, emphasizing their applications in summation, convergence analysis, and function approximation. Mastery of these topics enables deeper understanding of function behavior, convergence criteria, and practical applications in calculus.

References

  • Anton, H., Bivens, I., & Davis, S. (2016). Calculus: Early Transcendentals (11th ed.). John Wiley & Sons.
  • Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Brooks Cole.
  • Larson, R., & Edwards, B. (2013). Calculus (10th ed.). Brooks Cole.
  • Thomas, G. B., & Finney, R. L. (2002). Calculus and Analytic Geometry (9th ed.). Pearson.
  • Bressoud, D. M. (1994). A Radical Approach to Lebesgue Integration. Mathematical Association of America.
  • Spivak, M. (1994). Calculus. Publish or Perish.
  • Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
  • Lay, D. C. (2012). Linear Algebra and Its Applications (4th ed.). Pearson.
  • Gamelin, T. W. (2001). Complex Analysis. Springer.
  • Taylor, A. E. (1953). Fourier Series. Dover Publications.

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