Functions expressed as power series

Functions expressed as power series

In this section, we will learn how to convert functions into power series. Most of the functions we will be dealing with will be converted into a geometric series. After converting them into a power series, we will find the interval of convergence. Keep in mind that we do not have to check the endpoints of the inequality because we automatically know they will be divergent. Afterwards, we will look at an irregular function and express it as a power series. Integrals will be involved here. We will also take the derivative of a function and express that as a power series.
Basic concepts: Introduction to infinite series, Convergence & divergence of geometric series , Radius and interval of convergence with power series,

Lessons

Note *A formula that may be of use when expressing functions into power series:

11−r=∑n=0∞rn\frac{1}{1-r}=\sum_{n=0}^{\infty}r^n knowing that −1-1 < rr < 11

When finding the interval of convergence, there is no need to check the endpoints. This is because the sum of the geometric series strictly converges only when −1-1 < rr < 11, and not at r=1r=1.

If the function f(x)f(x) has a radius of convergence of RR, then the derivative and the anti-derivative of f(x)f(x) also has a radius of convergence of RR.
  • 1.
    Expressing Functions as Power Series
    Express the following functions as power series, and then find the interval of convergence:
    a)
    11−x2 \frac{1}{1-x^2}

    b)
    12−x \frac{1}{2-x}

    c)
    x21+3x2 \frac{x^2}{1+3x^2}

  • 2.
    Irregular Function expressed as a Power Series
    Express the function f(x)=ln(7−x)f(x)=ln(7-x) as a power series and find the interval of convergence.
  • 3.
    Derivative of a Function expressed as a Power Series
    Express the derivative of this function f(x)=11−xf(x)=\frac{1}{1-x} as a power series and find the interval of convergence.
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5.
Sequence and Series
5.1
Introduction to sequences
5.2
Monotonic and bounded sequences
5.3
Introduction to infinite series
5.4
Convergence and divergence of normal infinite series
5.5
Convergence & divergence of geometric series
5.6
Convergence & divergence of telescoping series
5.7
Divergence of harmonic series
5.8
P Series
5.9
Alternating series test
5.10
Divergence test
5.11
Comparison & limit comparison test
5.12
Integral test
5.13
Ratio test
5.14
Root test
5.15
Absolute & conditional convergence
5.16
Radius and interval of convergence with power series
5.17
Functions expressed as power series
5.18
Taylor series and Maclaurin series
5.19
Approximating functions with Taylor polynomials and error bounds
Calculus 2
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