What is Power series: Definition and 642 Discussions

In mathematics, a power series (in one variable) is an infinite series of the form

where an represents the coefficient of the nth term and c is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function.
In many situations c (the center of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form

Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewed as an example of a power series, with integer coefficients, but with the argument x fixed at 1⁄10. In number theory, the concept of p-adic numbers is also closely related to that of a power series.

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  1. E

    Power series vs. taylor series

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  2. J

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  3. I

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  4. C

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  5. T

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    I never took complex analysis in undergrad and always regretted it, so I'm working through the book Visual Complex Analysis on my own. Really enjoying it so far. Homework Statement Actually you can view the problem...
  6. H

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  7. V

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  8. L

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  9. J

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  10. H

    Questions about integration of power series

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  11. F

    Power Series Method: Solve y'' + 2xy' + 2y = 0

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  12. S

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  13. S

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  14. kreil

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  15. K

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  16. B

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  17. T

    Power series expansion of an exponential

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  18. D

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  19. L

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  20. G

    Using division of power series to find the first 3 nonzero terms

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  21. F

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  22. T

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  23. T

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  24. O

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  25. M

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  26. G

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  27. B

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  28. B

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  29. S

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  30. J

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  31. J

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  32. Hootenanny

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  33. R

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  35. F

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  36. T

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  37. A

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  38. R

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  39. R

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  40. J

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  41. F

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  42. M

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  43. M

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  44. M

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  45. rocomath

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  46. rocomath

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  47. R

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  48. S

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  49. G

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  50. A

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