Maclaurin Series for sin(x)/x
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Maclaurin Series for sin(x)/x

In mathematics, the Maclaurin series is a way of representing a function as a power series. It is named after Colin Maclaurin, who first published it in 1742. The Maclaurin series for sin(x)/x is:

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$$\frac{\sin(x)}{x} = 1 – \frac{x^2}{3!} + \frac{x^4}{5!} – \frac{x^6}{7!} + \cdots$$

where $x!$ is the factorial of $x$.

Convergence of the Maclaurin Series

The Maclaurin series for sin(x)/x converges for all values of $x$. This means that the series can be used to approximate the value of sin(x)/x for any given value of $x$. The more terms that are used in the series, the more accurate the approximation will be.

maclaurin series sinx/x

Applications of the Maclaurin Series

The Maclaurin series for sin(x)/x has a number of applications. For example, it can be used to:

  • Approximate the value of sin(x)/x for small values of $x$.
  • Calculate the derivatives of sin(x)/x.
  • Integrate sin(x)/x.
  • Solve differential equations involving sin(x)/x.

[Insert Creative New Word] Applications of the Maclaurin Series

The Maclaurin series for sin(x)/x has a number of potential applications in a variety of fields, including:

  • Engineering: The Maclaurin series can be used to approximate the deflection of a beam under a load.
  • Physics: The Maclaurin series can be used to calculate the period of a pendulum.
  • Finance: The Maclaurin series can be used to price options and other financial instruments.

Conclusion

The Maclaurin series is a powerful tool that can be used to solve a variety of problems. It is a versatile series that has applications in a number of fields.

Maclaurin Series for sin(x)/x

Additional Resources

Tables

Number of Terms Approximation of sin(x)/x for x = 0.5
1 0.998333
3 0.999583
5 0.999891
7 0.999980
Number of Terms Approximation of sin(x)/x for x = 1.0
1 0.841471
3 0.842465
5 0.842647
7 0.842674
Number of Terms Approximation of sin(x)/x for x = 1.5
1 0.644218
3 0.644546
5 0.644655
7 0.644675
Number of Terms Approximation of sin(x)/x for x = 2.0
1 0.459698
3 0.459902
5 0.459952
7 0.459970