Summing Maclaurin Series for x^2

Click For Summary
SUMMARY

The sum of the Maclaurin series for \( \sum \frac{x^{2k}}{k!} \) can be derived by substituting \( x^2 \) into the Taylor series expansion of \( e^x \). The original attempt to relate this series to known Taylor series such as \( \sin x \) and \( e^x \) was unsuccessful due to differences in factorial terms and the nature of the series' signs. Ultimately, the correct approach simplifies the problem significantly by leveraging the properties of the exponential function.

PREREQUISITES
  • Understanding of Maclaurin series
  • Familiarity with Taylor series expansions
  • Knowledge of factorial notation and its applications
  • Basic concepts of exponential functions
NEXT STEPS
  • Study the derivation of the Taylor series for \( e^x \)
  • Explore the properties of Maclaurin series
  • Investigate the relationship between factorials and series expansions
  • Learn about convergence of series and their applications in calculus
USEFUL FOR

Students studying calculus, mathematicians interested in series expansions, and educators teaching series convergence and function approximations.

JakeD
Messages
14
Reaction score
0

Homework Statement


How do I find the sum of \sum\frac{x^{2k}}{k!}?

The Attempt at a Solution


I tried transforming various known Taylor series, such as sin x, e^x, and so on, but they didn't fit for 2 reasons:
1. In all of them, the degree of the factor equals the power of x. i.e. if you have x^2k in the nominator, then you have (2k)! in the denominator, whereas here, you have x^2k in the nominator, while having k! (not (2k!)) in the denominator.

2. In sin x, you have alternating pluses and minuses, while in the required sum, they are all pluses.Any help will be appreciated
 
Physics news on Phys.org
OK, found the solution.

By replacing x with x^2 in the Taylor series of e^x, I get the desired sum.
 

Similar threads

Thermally insulated container with alternating series of walls
  • · Replies 49 ·
2
Replies
49
Views
3K
What is the meaning of the intercept of a particular solution to damped oscillator?
  • · Replies 7 ·
Replies
7
Views
1K
Graduate Coefficients of Chebyshev polynomials
  • · Replies 2 ·
Replies
2
Views
2K
Using known Maclaurin series to approximate modification of original
  • · Replies 3 ·
Replies
3
Views
2K
Taylor/Maclaurin series of a function
  • · Replies 6 ·
Replies
6
Views
2K
Understanding the generic nature of linearity in Physics
  • · Replies 2 ·
Replies
2
Views
1K
Interference pattern of a fan of plane waves
  • · Replies 2 ·
Replies
2
Views
2K
Approximate spring potential energy U(x) for small oscillations
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Another maclaurin vs. taylor series question
  • · Replies 3 ·
Replies
3
Views
2K
Is the Maclaurin series for sin(2x) the same as the one for sin(x)?
  • · Replies 2 ·
Replies
2
Views
3K